stochastic partial di erential equations: analysis and numerical...

Stochastic Partial Differential Equations:Analysis and Numerical Approximations

Arnulf Jentzen

September 14, 2015

2

Preface

These lecture notes have been written for the course “401-4606-00L Numerical Anal-ysis of Stochastic Partial Differential Equations” in the spring semester 2014 andin the spring semester 2015. These lecture notes are far away from being completeand remain under construction. In particular, these lecture notes do not yet con-tain a suitable comparison of the presented material with existing results, argumentsand notions in the literature. This will be the subject of a future version of theselecture notes. Furthermore, these lecture notes do not contain a number of proofs,arguments and intuitions. For most of this additional material, the reader is re-ferred to the lectures of the course “401-4606-00L Numerical Analysis of StochasticPartial Differential Equations” in the spring semester 2014. Sonja Cox and RyanKurniawan are gratefully acknowledged for their very helpful advice and assistance,especially for their help with the Matlab programs. Daniel Conus is also gratefullyacknowledged for several comments that helped to improve the presentation of theresults. In addition, we thank Antti Knowles for fruitful discussions on white noise.The students of the course “401-4606-00L Numerical Analysis of Stochastic PartialDifferential Equations” in the spring semester 2014 are gratefully acknowledged forpointing out a number of misprints to me. Special thanks are due to Timo Welti forbringing a number of misprints to my notice.

Zurich, February 2015

Arnulf Jentzen

3

Exercises

Solutions to the exercises can be turned in the designated mailbox in the anteroomHG G 53.x.

Exercise Exercises Deadlinesheet1 Exercises 1.1.8, 1.1.9, 2.2.6, and 2.4.4 05.03.2015, 10:15 AM2 Exercises 2.5.17, 2.5.20, 3.3.23, 3.5.2, and 3.5.3 19.03.2015, 10:15 AM3 Exercises 3.5.4, 3.5.15, 3.5.21, and 4.2.5 01.04.2015, 10:15 AM4 Exercises 4.3.4, 4.7.8, 5.3.22, and 6.1.6 15.04.2015, 10:15 AM5 Exercises 6.2.9, 6.2.11, 6.2.13, 6.2.14, and 6.2.21 24.04.2015, 10:15 AM6 Exercises 7.1.15, 8.1.14, and 8.1.15 07.05.2015, 10:15 AM7 Exercises 7.1.16, 7.2.2, and 8.2.5 14.05.2015, 10:15 AM8 Exercises 8.1.6, 9.1.6, and 9.4.1 21.05.2015, 10:15 AM

Contents

I Foundations in mathematical analysis 13

1 Gronwall-type inequalities 151.1 Properties of the beta and the gamma function . . . . . . . . . . . . 15

1.1.1 Functional equation of the gamma function . . . . . . . . . . . 161.1.2 Monotonicity properties of the gamma and the beta function . 161.1.3 Upper bounds for sums containing the beta and the gamma

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Integral operators related to the beta function . . . . . . . . . . . . . 191.3 Generalized exponential-type functions . . . . . . . . . . . . . . . . . 211.4 Generalized Gronwall-type inequalities . . . . . . . . . . . . . . . . . 21

1.4.1 Gronwall-type inequalities with a singularity at the initial time 241.4.2 Gronwall-type inequalities without a singularity at the initial

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Regularity of nonlinear functions 272.1 General functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Nonlinear characterization of the Borel sigma-algebra . . . . . 282.2.2 Pointwise limits of measurable functions . . . . . . . . . . . . 292.2.3 Lp-sets of measurable functions for p P r0,8q . . . . . . . . . . 31

2.3 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Strongly measurable functions . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Strongly measurable functions . . . . . . . . . . . . . . . . . . 332.4.3 Pointwise approximations of strongly measurable functions . . 342.4.4 Sums of strongly measurable functions . . . . . . . . . . . . . 362.4.5 Lp-spaces of strongly measurable functions for p P r0,8q . . . 37

2.5 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5

6 CONTENTS

2.5.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Semi-metric spaces . . . . . . . . . . . . . . . . . . . . . . . . 402.5.3 Continuity properties of functions . . . . . . . . . . . . . . . . 412.5.4 Modulus of continuity . . . . . . . . . . . . . . . . . . . . . . 422.5.5 Extensions of uniformly continuous functions . . . . . . . . . . 42

3 Linear functions 453.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 An intermezzo on sums over possibly uncountable index sets . . . . . 46

3.2.1 Fubini’s theorem in the case of non-sigma-finite measure spaces 463.2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2.1 Confinal sequences . . . . . . . . . . . . . . . . . . . 473.2.3 Sums over possibly uncountable index sets . . . . . . . . . . . 493.2.4 Fubini for sums . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 Best approximations and projections in Hilbert spaces . . . . 533.3.2 Examples of orthonormal bases . . . . . . . . . . . . . . . . . 54

3.3.2.1 Trigonometric functions . . . . . . . . . . . . . . . . 543.3.2.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq . . . . . . . . 553.3.2.3 Transformations of orthonormal bases . . . . . . . . 61

3.4 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.1 Continuous linear functions on normed vector spaces . . . . . 633.4.2 Compact operators on Banach spaces . . . . . . . . . . . . . . 653.4.3 Nuclear operators on Banach spaces . . . . . . . . . . . . . . . 65

3.4.3.1 Definition of Nuclear operators . . . . . . . . . . . . 653.4.3.2 Relation of bounded linear operators and nuclear op-

erators . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.3.3 Structure of the space of nuclear operators . . . . . . 683.4.3.4 Ideal property of the set of nuclear operators . . . . 693.4.3.5 Characterization of nuclear operators . . . . . . . . . 70

3.4.4 Hilbert-Schmidt operators on Hilbert spaces . . . . . . . . . . 703.4.4.1 Independence of the orthonormal basis . . . . . . . . 703.4.4.2 The Hilbert space of Hilbert-Schmidt operators . . . 713.4.4.3 Hilbert-Schmidt embeddings . . . . . . . . . . . . . . 72

3.5 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 733.5.1 Laplace operators on bounded domains . . . . . . . . . . . . . 74

3.5.1.1 Laplace operators with Dirichlet boundary conditions 753.5.1.2 Laplace operators with Neumann boundary conditions 76

CONTENTS 7

3.5.1.3 Laplace operators with periodic boundary conditions 773.5.2 Spectral decomposition for a diagonal linear operator . . . . . 783.5.3 Fractional powers of a diagonal linear operator . . . . . . . . . 803.5.4 Domain Hilbert space associated to a diagonal linear operator 813.5.5 Interpolation spaces associated to a diagonal linear operator . 82

3.6 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6.1 Existence and uniqueness of the Bochner integral . . . . . . . 843.6.2 Definition of the Bochner integral . . . . . . . . . . . . . . . . 85

4 Semigroups of bounded linear operators 874.1 Definition of a semigroup of bounded linear operators . . . . . . . . . 874.2 Types of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 The generator of a semigroup . . . . . . . . . . . . . . . . . . . . . . 884.4 A global a priori bound for semigroups . . . . . . . . . . . . . . . . . 904.5 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . 90

4.5.1 A priori bounds for strongly continuous semigroups . . . . . . 904.5.2 Pointwise convergence in the space of bounded linear operators 924.5.3 Existence of solutions of linear ordinary differential equations

in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.4 Domains of generators of strongly continuous semigroups . . . 944.5.5 Generators of strongly continuous semigroups . . . . . . . . . 954.5.6 A generalization of matrix exponentials to infinite dimensions 974.5.7 A characterization of strongly continuous semigroups . . . . . 98

4.6 Uniformly continuous semigroups . . . . . . . . . . . . . . . . . . . . 984.6.1 Matrix exponential in Banach spaces . . . . . . . . . . . . . . 994.6.2 Continuous invertibility of bounded linear operators in Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6.3 Generators of uniformly continuous semigroup . . . . . . . . . 1024.6.4 A characterization result for uniformly continuous semigroups 1034.6.5 An a priori bound for uniformly continuous semigroups . . . . 104

4.7 Semigroups generated by diagonal operators . . . . . . . . . . . . . . 1054.7.1 Semigroup generated by the Laplace operator . . . . . . . . . 1074.7.2 Smoothing effect of the semigroup . . . . . . . . . . . . . . . . 108

II Foundations in probability theory 111

5 Random variables with values in infinite dimensional spaces 1135.1 Borel sigma-algebras on normed vector spaces . . . . . . . . . . . . . 113

8 CONTENTS

5.1.1 The Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . 1135.1.2 Norm representations in normed vector spaces . . . . . . . . . 1145.1.3 Linear characterization of the Borel sigma-algebra . . . . . . . 115

5.2 Measures on normed vector spaces . . . . . . . . . . . . . . . . . . . . 1165.2.1 Uniqueness theorem for measures . . . . . . . . . . . . . . . . 1165.2.2 Fourier transform of a measure . . . . . . . . . . . . . . . . . 117

5.2.2.1 Characteristic functionals . . . . . . . . . . . . . . . 1175.2.2.2 Fourier transform on separable normed vector spaces 1185.2.2.3 Almost surely separably supported . . . . . . . . . . 1195.2.2.4 Trace set . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.2.5 Fourier transform on normed vector spaces . . . . . . 123

5.2.3 Covariance of a measure . . . . . . . . . . . . . . . . . . . . . 1255.2.3.1 The Baire category theorem on complete metric spaces1255.2.3.2 Regularities for correlations on normed vector spaces 1255.2.3.3 Covariances of measures and random variables . . . . 128

5.2.4 Gaussian measures on normed vector spaces . . . . . . . . . . 1295.2.4.1 Fourier transform of a Gaussian measure . . . . . . . 131

5.3 Probability measures on Hilbert spaces . . . . . . . . . . . . . . . . . 1325.3.1 Nuclear operators on Hilbert spaces . . . . . . . . . . . . . . . 1325.3.2 Expectation and covariance operator . . . . . . . . . . . . . . 1365.3.3 Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . 1385.3.4 Gaussian measures on Hilbert spaces . . . . . . . . . . . . . . 139

5.3.4.1 Karhunen-Loeve expansion . . . . . . . . . . . . . . 1395.3.4.2 Construction of Gaussian measures on Hilbert spaces 1405.3.4.3 Karhunen-Loeve expansion for Brownian motion . . 144

6 Stochastic processes 1536.1 Hilbert space valued stochastic processes . . . . . . . . . . . . . . . . 153

6.1.1 Standard Wiener processes . . . . . . . . . . . . . . . . . . . . 1536.1.2 Pseudo inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.1 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.2 Lenglart’s inequality . . . . . . . . . . . . . . . . . . . . . . . 1586.2.3 Modifications and indistinguishability . . . . . . . . . . . . . . 1626.2.4 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.2.5 Construction of the stochastic integral . . . . . . . . . . . . . 1656.2.6 Elementary processes revisited . . . . . . . . . . . . . . . . . . 1746.2.7 Cylindrical Wiener process . . . . . . . . . . . . . . . . . . . . 176

CONTENTS 9

III Stochastic Partial Differential Equations (SPDEs) 179

7 Solutions of SPDEs 1817.1 Existence, uniqueness and properties of mild solutions of SPDEs . . . 181

7.1.1 Mild solutions of SPDEs . . . . . . . . . . . . . . . . . . . . . 1817.1.2 A setting for SPDEs with globally Lipschitz continuous non-

linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.1.3 A strong perturbation estimate for SPDEs . . . . . . . . . . . 1847.1.4 Uniqueness of mild solutions of SPDEs . . . . . . . . . . . . . 188

7.1.4.1 Uniqueness of predictable mild solutions of SEEs withglobally Lipschitz continuous coefficients . . . . . . . 188

7.1.4.2 Uniqueness of left-continuous mild solutions of SEEswith semi-globally Lipschitz continuous coefficients . 188

7.1.5 Existence and regularity of mild solutions of SPDEs . . . . . . 1927.1.6 A priori bounds for mild solutions of SPDEs . . . . . . . . . . 1927.1.7 Temporal-regularity of solution processes of SPDEs . . . . . . 1977.1.8 Existence of continuous solutions . . . . . . . . . . . . . . . . 198

7.2 Examples of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.2.1 Second order SPDEs . . . . . . . . . . . . . . . . . . . . . . . 200

IV Numerical Analysis of SPDEs 205

8 Strong numerical approximations for SPDEs 2078.1 Spatial spectral Galerkin approximations for SPDEs . . . . . . . . . . 207

8.1.1 Galerkin projections . . . . . . . . . . . . . . . . . . . . . . . 2078.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148.1.3 A strong numerical approximation result for spectral Galerkin

approximations of SPDEs . . . . . . . . . . . . . . . . . . . . 2148.2 Temporal numerical approximations for SPDEs . . . . . . . . . . . . 218

8.2.1 Euler type approximations for SPDEs . . . . . . . . . . . . . . 2198.2.1.1 Exponential Euler method . . . . . . . . . . . . . . . 2198.2.1.2 Accelerated exponential Euler method . . . . . . . . 2208.2.1.3 Linear-implicit Euler method . . . . . . . . . . . . . 2218.2.1.4 Linear-implicit Crank-Nicolson-Euler method . . . . 222

8.2.2 Nonlinearity-stopped Euler type approximations for SPDEs . . 2238.2.2.1 Nonlinearity-stopped exponential Euler method . . . 2248.2.2.2 Nonlinearity-stopped linear-implicit Euler method . . 225

8.2.3 Milstein type approximations for SPDEs . . . . . . . . . . . . 226

10 CONTENTS

8.2.3.1 Exponential Milstein method . . . . . . . . . . . . . 2268.2.3.2 Linear-implicit Milstein method . . . . . . . . . . . . 2288.2.3.3 Linear-implicit Crank-Nicolson-Milstein method . . . 229

8.2.4 Strong convergence analysis for exponential Euler approxima-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

8.3 Noise approximations for SPDEs . . . . . . . . . . . . . . . . . . . . 2418.3.1 Noise perturbation estimates . . . . . . . . . . . . . . . . . . . 2418.3.2 Noise approximations for SPDEs . . . . . . . . . . . . . . . . 242

8.4 Full discretizations for SPDEs . . . . . . . . . . . . . . . . . . . . . . 2458.4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.4.2 Full-discrete spectral Galerkin exponential Euler method for

SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.4.3 Full-discrete spectral Galerkin linear-implicit Euler method for

SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2488.4.4 Full-discrete spectral Galerkin nonlinearity-stopped exponen-

tial Euler method for SPDEs . . . . . . . . . . . . . . . . . . . 2508.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicit

Euler method for SPDEs . . . . . . . . . . . . . . . . . . . . . 251

9 Weak numerical approximations for SPDEs 2559.1 An Ito type formula for SPDEs . . . . . . . . . . . . . . . . . . . . . 255

9.1.1 A setting for mild stochastic calculus . . . . . . . . . . . . . . 2559.1.2 Mild stochastic processes . . . . . . . . . . . . . . . . . . . . . 2569.1.3 Mild Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . 257

9.2 Solution processes of SPDEs . . . . . . . . . . . . . . . . . . . . . . . 2609.3 Transformations of semigroups of solutions of SPDEs . . . . . . . . . 2619.4 Weak convergence for temporal numerical approximations for SPDEs 2639.5 Weak convergence of Galerkin projections for SPDEs . . . . . . . . . 263

9.5.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.5.2 Weak convergence for spatial spectral Galerkin projections . . 264

10 Additional material 26910.1 Egorov’s theorem on almost uniform convergence . . . . . . . . . . . 269

10.1.1 General measure spaces . . . . . . . . . . . . . . . . . . . . . 26910.1.1.1 Almost sure convergence . . . . . . . . . . . . . . . . 26910.1.1.2 Luzin uniform-type convergence . . . . . . . . . . . . 27110.1.1.3 Almost uniform convergence . . . . . . . . . . . . . . 271

10.1.2 Finite measure spaces . . . . . . . . . . . . . . . . . . . . . . 272

CONTENTS 11

10.1.3 Sigma-finite measure spaces . . . . . . . . . . . . . . . . . . . 27310.1.4 General measure spaces . . . . . . . . . . . . . . . . . . . . . 274

10.2 Fast convergence in probability . . . . . . . . . . . . . . . . . . . . . 27410.3 Dini’s theorem on pointwise convergence of continuous functions . . . 275

10.3.1 On the compactness of the argument space . . . . . . . . . . . 27710.3.2 On the monotonicity of the approximating functions . . . . . . 27710.3.3 On the continuity of the approximating functions . . . . . . . 27810.3.4 On the continuity of the limit function . . . . . . . . . . . . . 279

11 Solutions to selected exercises 28111.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

11.1.1 Solution to Exercise 2.2.6 . . . . . . . . . . . . . . . . . . . . 28111.1.2 Solution to Exercise 2.4.4 . . . . . . . . . . . . . . . . . . . . 281

11.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28211.2.1 Solution to Exercise 3.5.2 . . . . . . . . . . . . . . . . . . . . 282

11.3 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.3.1 Solution to Exercise 7.1.15 . . . . . . . . . . . . . . . . . . . . 28311.3.2 Solution to Exercise 7.1.16 . . . . . . . . . . . . . . . . . . . . 28511.3.3 Solution to Exercise 7.2.2 . . . . . . . . . . . . . . . . . . . . 286

11.4 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.4.1 Solution to Exercise 8.1.6 . . . . . . . . . . . . . . . . . . . . 28611.4.2 Solution to Exercise 8.1.15 . . . . . . . . . . . . . . . . . . . . 28811.4.3 Solution to Exercise 8.2.5 . . . . . . . . . . . . . . . . . . . . 291

11.5 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29211.5.1 Solution to Exercise 9.4.1 . . . . . . . . . . . . . . . . . . . . 292

12 CONTENTS

Part I

Foundations inmathematical analysis

13

Chapter 1

Gronwall-type inequalities

This chapter is based on Section 7.1 in Henry [10].

1.1 Properties of the beta and the gamma func-

tion

For completeness we first recall the definition of the beta function and the gammafunction.

Definition 1.1.1 (Beta function and gamma function). We denote by B : p0,8q2 Ñp0,8q and Γ: p0,8q Ñ p0,8q the functions with the property that for all x, y P p0,8qit holds that

Bpx, yq “

ż 1

0

tpx´1qp1´ tqpy´1q dt (1.1)

and

Γpxq “

ż 8

0

tpx´1q e´t dt, (1.2)

we call B the beta function, and we call Γ the gamma function.

15

16 CHAPTER 1. GRONWALL-TYPE INEQUALITIES

1.1.1 Functional equation of the gamma function

Lemma 1.1.2 (Basic properties of the gamma function and the Beta function). Forall x, y P p0,8q, n P N0 it holds that

Bpx, yq “ Bpy, xq “Γpxq ¨ Γpyq

Γpx` yq“

ż 8

0

tpx´1q

p1` tqpx`yqdt, (1.3)

Γpn` 1q “ n! and Γpx` 1q “ x ¨ Γpxq . (1.4)

Proof of Lemma 1.1.2. First, observe that the integral transformation theorem en-sures that for all x, y P p0,8q it holds that

Bpx, yq “

ż 1

0

tpx´1qp1´ tqpy´1q dt “

ż 8

1

“

1t

‰px´1q “1´ 1

t

‰py´1q 1t2dt

“

ż 8

1

tp´x´1q“

t´1t

‰py´1qdt “

ż 8

1

tp´xýq pt´ 1qpy´1q dt

“

ż 8

0

pt` 1qp´xýq tpy´1qdt “

ż 8

0

tpy´1q

pt` 1qpx`yqdt.

(1.5)

Moreover, note that for all x P p0,8q it holds that

Γpx` 1q “

ż 8

0

tppx`1q´1q e´t dt “ ´

ż 8

0

tx“

é´t‰

dt

“ ´

ˆ

“

txe´t‰t“8

t“0´ x

ż 8

0

tpx´1q e´t dt

˙

“ x

ż 8

0

tpx´1q e´t dt “ x ¨ Γpxq.

(1.6)

The proof of Lemma 1.1.2 is thus completed.

1.1.2 Monotonicity properties of the gamma and the betafunction

Lemma 1.1.3 (Montonicity property of the gamma function). It holds that

limxÑ8

Γ1pxq “ 8 (1.7)

and there exists a real number C P p0,8q such that for all x, y P rC,8q with x ď yit holds that Γpxq ď Γpyq.

1.1. PROPERTIES OF THE BETA AND THE GAMMA FUNCTION 17

Proof of Lemma 1.1.3. Observe that for all x P p0,8q it holds that

Γ1pxq “

ż 8

0

lnptq tpx´1q e´t dt

“

ż 1

0

lnptq tpx´1q e´t dt`

ż e

1

lnptq tpx´1q e´t dt`

ż 8

e


ě inftPp0,1q

“

lnptq tpx´1q e´t‰

`

ż 8

e


ě inftPp0,1q

“

lnptq tpx´1q‰

`

ż 8

e

tpx´1q e´t dt.

(1.8)

This proves that

limxÑ8

Γ1pxq ě inftPp0,1q

rlnptq ts ` limxÑ8

ż 8

e

tpx´1q e´t dt “ 8. (1.9)


Lemma 1.1.4 (Monotonicity of the beta function). For all x, y, x, y P p0,8q withx ď x and y ď y it holds that

Bpx, yq ď Bpx, yq. (1.10)

Proof of Lemma 1.1.4. Note that for all θ P p0, 1q, x, x P R with x ď x it holds that

θx ď θx. (1.11)

Combining this with Definition 1.1.1 completes the proof of Lemma 1.1.4.

1.1.3 Upper bounds for sums containing the beta and thegamma function

Lemma 1.1.5 (An upper bound for the beta function). Let x, y P p0,8q withpx´ 1q py ´ 1q ě 0. Then

Bpx, yq “

ż 1

0

p1´ tqpx´1q tpy´1q dt ď

ż 1

0

tpx`y´2q dt “

#

8 : x` y ď 11

px`y´1q: x` y ą 1

. (1.12)


Proof of Lemma 1.1.5. First, observe that the equalities in (1.12) are clear. It thusremains to prove the inequality in (1.12). For this we assume w.l.o.g. that x` y ą 1,that x ‰ 1 and that y ‰ 1 (otherwise also the inequality in (1.12) is clear). Theassumption that px´ 1q py ´ 1q ě 0 hence shows that px´ 1q py ´ 1q ą 0 and thatpy´1qpx´1q

P p0,8q. Combining this with Holder’s inequality proves that

ż 1

0

p1´ tqpx´1q tpy´1q dt

ď

„ż 1

0

p1´ tqpx´1qr1`py´1qpx´1qs dt

´

r1`py´1qpx´1qs

´1¯

„ż 1

0

tpy´1qr1`px´1qpy´1q s dt

´

r1`px´1qpy´1q s

´1¯

“

„ż 1

0

p1´ tqpx`y´2q dt

´

r1`py´1qpx´1qs

´1¯

„ż 1

0

tpx`y´2q dt

´

r1`px´1qpy´1q s

´1¯

“

ż 1

0

tpx`y´2q dt.

(1.13)


Remark 1.1.6. Lemma 1.1.5, in particular, shows that for all x, y P p0,8q withpx´ 1q py ´ 1q ě 0 it holds that

Bpx, yq ď

ż 1

0

tpx`y´2q dt. (1.14)

However, it is not true that for all x, y P p0,8q it holds that

Bpx, yq ď

ż 1

0

tpx`y´2q dt. (1.15)

Indeed, observe that

limxŒ0

Bpx, 2´ xq “ limxŒ0

ż 1

0

p1´ tqpx´1q tp2´xq dt ě limxŒ0

ż 1

12

p1´ tqpx´1q tp2´xq dt

ě

„

1

2

p2´xq

limxŒ0

ż 1

12

p1´ tqpx´1q dt “ 8 ą 1 “

ż 1

0

tp2´2q dt.

(1.16)

Exercise 1.1.7. Prove that for all c P r0,8q, ε P p0,8q it holds that

8ÿ

n“1

cn

Γpnεqă 8. (1.17)

1.2. INTEGRAL OPERATORS RELATED TO THE BETA FUNCTION 19

Exercise 1.1.8. Prove that for all c P r0,8q, ε P p0,8q it holds that

8ÿ

n“1

cn„

nś

k“1

B`

kε, ε˘

ă 8. (1.18)

Exercise 1.1.9. Prove that for all c P r0,8q, ε, δ, ρ P p0,8q it holds that

8ÿ

n“1

cn„

n´1ś

k“0

B`

ε` kδ, ρ˘

ă 8. (1.19)

1.2 Integral operators related to the beta function

Lemma 1.2.1 (A scaling property of the beta function). For all β, γ P p0,8q,r, t P r0,8q with r ď t it holds that

ż t

r

pt´ sqpβ´1qps´ rqpγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq. (1.20)

Proof of Lemma 1.2.1. Note that for all β, γ P p0,8q, r, t P r0,8q with r ď t it holdsthat

ż t

r

pt´ sqpβ´1qps´ rqpγ´1q ds “

ż pt´rq

0

pt´ r ´ sqpβ´1q spγ´1q ds

“ pt´ rqpβ`γ´1q

ż 1

0

p1´ sqpβ´1q spγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq.

(1.21)


The next estimate, inequality (1.22) in Lemma 1.2.2, is an immediate consequenceof Lemma 1.2.1.

Lemma 1.2.2. Let α, γ, τ P R, T P rτ,8q, u PMpBprτ, T sq,Bpr0,8sqq, β P p0,8qsatisfy α ` γ ą 1. Then it holds for all t P rτ, T s that

ż t

τ

pt´ sqpβ´1qps´ τqpγ´1q upsq ds

ď pt´ τqpα`β`γ´2qB`

β, α ` γ ´ 1˘

«

supsPpτ,tq

upsq

ps´ τqpα´1q

ff

.

(1.22)


We need a further estimate for the integral operator appearing on the left handside of (1.22). This is the subject of the next lemma.

Lemma 1.2.3 (Iterations of an integral operator). Let α, γ, τ P R, T P rτ,8q, b Pr0,8q, β P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq Ñ MpBprτ, T sq,Bpr0,8sqq, assumethat mintα, βu`γ ą 1, and assume that for all u PMpBprτ, T sq,Bpr0,8sqq, t P rτ, T sit holds that

`

Bpuq˘

ptq “ b

ż t

τ

pt´ sqpβ´1qps´ τqpγ´1q upsq ds. (1.23)

Then it holds for all n P N, t P rτ, T s, u PMpBprτ, T sq,Bpr0,8sqq that`

Bnpuq

˘

ptq

ď bn pt´ τqpα´1`npβ`γ´1qq

«

n´1ź

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ff«

supsPpτ,tq

upsq

ps´τqpα´1q

ff

(1.24)

and that

`

Bnpuq

˘

ptq ď bn pt´ τqpn´1qrγ´1s`

«

n´1ź

k“1

B`

β, kpβ ´ r1´ γs`q˘

ff

¨t

∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q

ps´ τqpγ´1q upsq ds.

(1.25)

Proof of Lemma 1.2.3. Estimate (1.24) is an immediate consequence of Lemma 1.2.2.It thus remains to prove estimate (1.25). For this we assume in the following w.l.o.g.that τ “ 0. Then note that Lemma 1.2.1 implies that for all u PMpBpr0, T sq,Bpr0,8sq,t P r0, T s, b P r0,8q, β, γ P p0,8q with β ` γ ą 1 it holds that

b

ż t

0

pt´ sqpβ´1q spγ´1q

„

b

ż s

0

ps´ rqpβ´1q rpγ´1q uprq dr

ds

“ b b

ż t

0

ż s

0

pt´ sqpβ´1q spγ´1qps´ rqpβ´1q rpγ´1q uprq dr ds

“ b b

ż t

0

rpγ´1q uprq

„ż t

r

pt´ sqpβ´1q spγ´1qps´ rqpβ´1q ds

dr

ď b b trγ´1s`ż t

0

rpγ´1q uprq

„ż t

r

pt´ sqpβ´1qps´ rqpβ´1`mintγ´1,0uq ds

dr

ď B`

β, β ´ r1´ γs`˘

b b trγ´1s`ż t

0

pt´ rqpβ`β´r1´γs`´1q rpγ´1q uprq dr.

(1.26)

1.3. GENERALIZED EXPONENTIAL-TYPE FUNCTIONS 21

Iterating (1.26) shows that for all u PMpBpr0, T sq,Bpr0,8sq, t P r0, T s, n P t2, 3, . . . uit holds that

`

Bnpuq

˘

ptq ď bn tpn´1qrγ´1s`t

∫0pt´ sqpβ`pn´1qpβ´r1´γs`q´1q spγ´1q upsq ds

¨

«

n´1ź

k“1

B`


ff

.

(1.27)


1.3 Generalized exponential-type functions

Definition 1.3.1 (Generalized exponential-type functions). We denote by Er : r0,8q Ñr0,8q, r P p0,8q, Er : r0,8q Ñ r0,8q, r P p0,8q, and Er : r0,8q Ñ r0,8q,r P p0,8q, the functions with the property that for all r P p0,8q, x P r0,8q itholds that

Errxs “8ÿ

n“0

xnr

Γpnr ` 1q, Errxs “ Er

”

pxΓprqq1r

ı

“

8ÿ

n“0

pxΓprqqn

Γpnr ` 1q(1.28)

and Errxs “b

Errx2s “

«

8ÿ

n“0

px2Γprqqn

Γpnr ` 1q

ff12

. (1.29)

1.4 Generalized Gronwall-type inequalities

Lemma 1.4.1 (Main idea in the proof of the generalized Gronwall inequality). Letτ P R, T P rτ,8q, b P MpBprτ, T s2q,Bpr0,8sqq, a, e P MpBprτ, T sq,Bpr0,8sqq,B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy that for all t P rτ, T s,u PMpBprτ, T sq,Bpr0,8sqq it holds that

`

Bpuq˘

ptq “t

∫τbpt, squpsq ds, (1.30)

and assume that e ď a`Bpeq. Then it holds for all n P N that

e ďn´1ÿ

k“0

Bkpaq `Bn

peq. (1.31)


Proof of Lemma 1.4.1. Estimate (1.31) follows immediately from an iterated applica-tion of the assumption e ď a`Bpeq and from the fact that B is monotone in the sensethat for all u, u P MpBprτ, T sq,Bpr0,8sqq with u ď u it holds that Bpuq ď Bpuq.The proof of Lemma 1.4.1 is thus completed.

Next we present the generalized Gronwall inequalities. They are modified versionsof the estimates in Section 7.1 in Henry [10].

Theorem 1.4.2. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β, γ P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy β ` γ ą

1 andşT

τps´ τqpγ´1q epsq ds ă 8, assume that for all u P MpBprτ, T sq,Bpr0,8sqq,

t P rτ, T s it holds that

pBuqptq “ bt

∫τpt´ sqpβ´1q

ps´ τqpγ´1q upsq ds, (1.32)

and assume that for all t P rτ, T s it holds that

eptq ď aptq ` bt

∫τpt´ sqpβ´1q

ps´ τqpγ´1q epsq ds. (1.33)

Then it holds for all t P rτ, T s that

eptq ď8ÿ

n“0

`

Bnpaq

˘

ptq ď aptq `8ÿ

n“1

bn pt´ τqpn´1qrγ´1s`„

n´1ś

k“1

B`


¨t

∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q

ps´ τqpγ´1q apsq ds (1.34)

and it holds for all t P pτ, T s, α P p0,8q with α ` γ ą 1 that

eptq ď8ÿ

n“0

`

Bnpaq

˘

ptq ď aptq (1.35)

`

«

supsPpτ,tq

apsq

ps´τqpα´1q

ff

8ÿ

n“1

bn pt´ τqpα´1`npβ`γ´1qq

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

looooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooon

ă8

.

Proof of Theorem 1.4.2. W.l.o.g. we assume that τ “ 0. Lemma 1.4.1 implies thatfor all n P N0 it holds that

e ď a`Bpaq `B2paq ` . . .`Bn

paq `Bpn`1qpeq “

«

nÿ

k“0

Bkpaq

ff

`Bpn`1qpeq. (1.36)

1.4. GENERALIZED GRONWALL-TYPE INEQUALITIES 23

Next we note that inequality (1.25) in Lemma 1.2.3 together with the assumptionthat @ t P r0, T s :

şt

0spγ´1q epsq ds ă 8 and the fact that

@ c P r0,8q, r P p0,8q : limnÑ8

„

cn

Γppn´ 1qrq

“ 0 (1.37)

(see Exercise 1.1.7) implies that for all t P p0, T s it holds that

limnÑ8

`

Bnpeq

˘

ptq

ď limnÑ8

«

bn tβ´1`pn´1qpβ`γ´1qt

∫0spγ´1qepsq ds

«

n´1ź

k“1

B`


ffff

ď∫ t0 spγ´1q epsq ds

tγlimnÑ8

«

“

tpβ`γ´1q b‰n

«

n´1ź

k“1

B`


ffff


tγlimnÑ8

«

“

maxp1,Γpβqq tpβ`γ´1q b‰n

«

n´1ź

k“1

Γpkpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

ffff


tγ

¨ limnÑ8

«

“

|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n

Γ`

β ` pn´ 1qpβ ´ r1´ γs`q˘

„

n´2ś

k“1

Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

ff


tγ

„

8ś

k“1

Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq

¨ limnÑ8

«

“

|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n

Γ`

pn´ 1qpβ ´ r1´ γs`q˘

ff

“ 0.

(1.38)

This and (1.36) prove the first inequalities in (1.34) and (1.35). Estimate (1.25) inLemma 1.2.3 proves the second inequality in (1.34). Furthermore, estimate (1.24)in Lemma 1.2.3 proves the second inequality in (1.35). Finally, observe that Exer-cise 1.1.9 implies that for all t P pτ, T s it holds that

8ÿ

n“1

bn pt´ τqnpβ`γ´1q

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

“

8ÿ

n“1

”

b pt´ τqpβ`γ´1qın

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ă 8.

(1.39)

The proof of Theorem 1.4.2 is thus completed.


1.4.1 Gronwall-type inequalities with a singularity at the ini-tial time

The next result, Corollary 1.4.3, specialises estimate (1.34) in Theorem 1.4.2 to thecase where γ “ 1; see Lemma 7.1.1 in Henry [10].

Corollary 1.4.3. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β P p0,8q satisfy

şT

τepsq ds ă 8 and assume that for all t P rτ, T s it holds that

eptq ď aptq ` bt

∫τpt´ sqpβ´1q epsq ds. (1.40)


eptq ď aptq ` rΓpβq bs1β

ż t

τ

E1β“

pt´ sq rΓpβq bs1β‰

apsq ds. (1.41)

Proof of Corollary 1.4.3. Inequality (1.34) in Theorem 1.4.2 with γ “ 1 shows thatfor all t P rτ, T s it holds that

eptq ď aptq `

ż t

τ

«

8ÿ

n“1

rΓpβq bsn pt´ sqpnβ´1q

Γpnβq

ff

apsq ds

“ aptq ` rΓpβq bs1β

ż t

τ

«

8ÿ

n“1

“

pt´ sq rΓpβq bs1β‰pnβ´1q

Γpnβq

ff

apsq ds.

(1.42)

Next note that Lemma 1.1.2 shows that for all x P p0,8q it holds that

E1βpxq “8ÿ

n“1

nβxpnβ´1q

Γpnβ ` 1q“

8ÿ

n“1

xpnβ´1q

Γpnβq. (1.43)

Combining this with (1.42) completes the proof of Corollary 1.4.3.

The next result, Corollary 1.4.4, specialises estimate (1.35) in Theorem 1.4.2 tothe case where the function a in Theorem 1.4.2 satisfies aptq “ c tpα´1q for all t P pτ, T sand some c P r0,8q; see Exercise 3 in Henry [10]. Corollary 1.4.4 follows immediatelyfrom (1.35) in Theorem 1.4.2 and from Exercise 1.1.9.

1.4. GENERALIZED GRONWALL-TYPE INEQUALITIES 25

Corollary 1.4.4. Let τ P R, a, b P r0,8q, T P rτ,8q, α, β, γ P p0,8q, a, e P

MpBprτ, T sq,Bpr0,8sqq satisfy mintα, βu ` γ ą 1 andşT

τps´ τqpγ´1q epsq ds ă 8

and assume that for all t P pτ, T s it holds that

eptq ď a pt´ τqpα´1q` b

t

∫τpt´ sqpβ´1q

ps´ τqpγ´1q epsq ds. (1.44)

Then it holds for all t P pτ, T s that

eptq ď a pt´ τqpα´1q8ÿ

n“0

bn pt´ τqnpβ`γ´1q

„

n´1ś

k“0

B`

β, α ` γ ´ 1` kpβ ` γ ´ 1q˘

ă 8.

(1.45)

The next result, Corollary 1.4.5, specialises Corollary 1.4.4 to the case γ “ 1; cf.Exercise 4 in Henry [10]. Corollary 1.4.5 follows immediately from Corollary 1.4.4.

Corollary 1.4.5. Let τ P R, T P rτ,8q, e PMpBprτ, T sq,Bpr0,8sqq, a, b P r0,8q,α, β P p0,8q satisfy

şT

τepsq ds ă 8 and assume that for all t P pτ, T s it holds that

eptq ď a pt´ τqpα´1q` b

t


Then it holds for all t P pτ, T s that

eptq ď a pt´ τqpα´1q8ÿ

n“0

bn pt´ τqnβ„

n´1ś

k“0

B`

β, α ` kβ˘

ă 8. (1.47)

1.4.2 Gronwall-type inequalities without a singularity at theinitial time

In the remainder of these lecture notes we will often use the following Gronwall-typeestimate; see Lemma 7.1.1 in Henry [10].

Corollary 1.4.6. Let τ P R, β P p0,8q, a, b P r0,8q, e PMpBprτ, T sq,Bpr0,8sqqsatisfy

şT

τepsq ds ă 8 and assume that for all t P rτ, T s it holds that

eptq ď a` bt



eptq ď a ¨ Eβ

”

pt´ τq pbΓpβqq1βı

“ a ¨ Eβ

“

pt´ τqβ b‰

. (1.49)


Proof of Corollary 1.4.6. Corollary 1.4.3 implies that for all t P rτ, T s it holds that

eptq ď a` rΓpβq bs1β

ż t

τ

E1β“

pt´ sq rΓpβq bs1β‰

a ds. (1.50)

The fundamental theorem of calculus hence shows that for all t P rτ, T s it holds that

eptq ď a

„

1` limεŒ0

ż t

τ`ε

rΓpβq bs1β E1β

“

ps´ τq rΓpβq bs1β‰

ds

“ a

„

1` limεŒ0

“

Eβ

“

pt´ τq rΓpβq bs1β‰

´ Eβ

“

ε rΓpβq bs1β‰‰

“ a“

1``

Eβ

“


´ Eβ

“

0‰˘‰

“ aEβ

“


.

(1.51)

The proof of Corollary 1.4.6 is thus completed.

Chapter 2

Regularity of nonlinear functions

Most of this chapter is based on Da Prato & Zabczyk [7] and Prevot & Rockner [24].

2.1 General functions

Definition 2.1.1 (Power set). Let A be a set. Then we denote by PpAq the powerset of A (the set of all subsets of A).

Definition 2.1.2 (Set of functions). Let A and B be sets. Then we denote byMpA,Bq the set of all functions from A to B.

Definition 2.1.3 (Domain of definition, range/codomain, image of a function). LetA and B be sets and let f : AÑ B be a function. Then

• we denote by Dpfq the set given by Dpfq “ A and we call Dpfq the domain ofdefinition of f ,

• we denote by rangepfq the set given by rangepfq “ B and we call rangepfq therange/codomain of f ,

• we denote by impfq the set given by impfq “ fpAq and we call impfq the imageof f , and

• we denote by graphpfq the set given by graphpfq “ tpx, yq P AˆB : y “ fpxquand we call graphpfq the graph of f .

27

28 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS

2.2 Measurable functions

We first recall the notion of a measurable mapping.

Definition 2.2.1 (Measurable functions). Let pΩ1,F1q and pΩ2,F2q be measurablespaces and let f : Ω1 Ñ Ω2 be a function with the property that for all F P F2 it holdsthat

f´1pF q P F1. (2.1)

Then f is called F1/F2-measurable.

Definition 2.2.2 (Measurable functions). Let pΩ1,F1q and pΩ2,F2q be measurablespaces. Then we denote by MpF1,F2q the set of all F1/F2-measurable functions.

Definition 2.2.3 (Borel sigma-algebra). Let pE, Eq be a topological space. Then wedenote by BpEq the set given by BpEq “ σEpEq and we call BpEq the Borel sigma-algebra on pE, Eq.

Definition 2.2.4 (Distance of sets). Let pE, dEq be a metric space. Then we denoteby distE : PpEq ˆ PpEq Ñ r0,8s the function with the property that for all A,B P

PpEq it holds that

distEpA,Bq “

#

infaPA infbPB dEpa, bq : A ‰ H and B ‰ H

8 : else. (2.2)

2.2.1 Nonlinear characterization of the Borel sigma-algebra

The next proposition shows that if pE, dEq is a metric space, then BpEq is the smallestsigma-algebra with the property that every continuous real-valued function is Borelmeasurable.

Proposition 2.2.5 (Nonlinear characterization of the Borel sigma-algebra). LetpE, dEq be a metric space. Then

BpEq “ σE`

pϕqϕPCpE,Rq˘

“ σEpϕ : ϕ P CpE,Rqq

“ σE`

ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq

(˘

.(2.3)

2.2. MEASURABLE FUNCTIONS 29

Proof of Proposition 2.2.5. First of all, observe that for every ϕ P CpE,Rq it holdsthat ϕ is BpEq/BpRq-measurable. Hence, we obtain that

BpEq Ě σE`


(˘

. (2.4)

It thus remains to prove that

BpEq Ď σE`


(˘

. (2.5)

For this observe by that definition it holds that

BpEq “ σEptA P PpEq : A is an open set in pE, dEquq . (2.6)


tA P PpEq : A is an open set in pE, dEqu

Ď σE`


(˘

.(2.7)

For this let B Ă E be an open set in pE, dEq and let ψ : E Ñ R be the function withthe property that for all x P E it holds that

ψpxq “ distEptxu, EzBq. (2.8)

Then it holds that ψ P CpE,Rq and this implies that

B “ ψ´1pp0,8qq Ď σE

`


(˘

. (2.9)

The proof of Proposition 2.2.5 is thus completed.

Exercise 2.2.6. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space, andlet f : Ω Ñ E be a function. Prove that f is F/BpEq-measurable if and only if itholds for all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.

2.2.2 Pointwise limits of measurable functions

Definition 2.2.7 (Sets of numbers). We denote by

N “ t1, 2, 3, . . . u (2.10)

the set of natural numbers, we denote by

N0 “ NY t0u “ t0, 1, 2, . . . u (2.11)

the union of t0u and the set of natural numbers, we denote by

Z “ t0, 1,´1, 2,´2, . . . u (2.12)

the set of integer numbers, we denote by Q the set of rational numbers, we denoteby R the set of real numbers and we denote by C the set of complex numbers.


Note thatN Ă N0 Ă Z Ă Q Ă R Ă C. (2.13)

Lemma 2.2.8. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function, andlet Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings such that forall ω P Ω it holds that Y pωq “ supnPNXnpωq. Then Y is F/BpRq-measurable.

Proof of Lemma 2.2.8. Note that for all c P R it holds that

tY ď cu “

"

supnPN

Xn ď c

*

“č

nPN

tXn ď culoooomoooon

PF

P F . (2.14)


Lemma 2.2.9. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function, andlet Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings such that forall ω P Ω it holds that limnÑ8Xnpωq “ Y pωq. Then Y is F/BpRq-measurable.

Proof of Lemma 2.2.9. Note that Lemma 2.2.8 implies that for all c P R it holdsthat

tY ě cu “!

limnÑ8

Xn ě c)

“

"

lim supnÑ8

Xn ě c

*

“

#

limnÑ8

«

supmPtn,n`1,... u

Xm

ff

ě c

+

“č

nPN

#«

supmPtn,n`1,... u

Xm

ff

ě c

+

looooooooooooooomooooooooooooooon

PF

P F . (2.15)


The next corollary is an immediate consequence of Exercise 2.2.6 and Lemma 2.2.9;see, e.g., Proposition E.1 in [2] and Proposition A.1.3 in Prevot & Rockner [24].

Corollary 2.2.10. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space,and let f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if thereexists a sequence gn : Ω Ñ E, n P N, of F/BpEq-measurable functions such that forall ω P Ω it holds that

limnÑ8

dEpfpωq, gnpωqq “ 0. (2.16)

2.2. MEASURABLE FUNCTIONS 31

2.2.3 Lp-sets of measurable functions for p P r0,8q

Definition 2.2.11 (Lp-sets for p P r0,8q). Let K P tR,Cu, let pΩ,A, µq be ameasure space, let q P p0,8q, and let pV, ¨V q be a normed K-vector space. Then we

denote by L0pµ; ¨V q the set given by

L0pµ; ¨V q “MpA,BpV qq, (2.17)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that

for all f P L0pµ; ¨V q it holds that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

, (2.18)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (2.19)

Definition 2.2.12 (Lp-sets for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Then

we denote by r¨sL0pµ;¨V q: L0pµ; ¨V q Ñ P

`

L0pµ; ¨V q˘

the function with the property

that for all f P L0pµ; ¨V q it holds that

rf sL0pµ;¨V q“

!

g P L0pµ; ¨V q : pDA P F : µpAq “ 0 and tf ‰ gu Ď Aq

)

, (2.20)

we denote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “

rf sLppµ;¨V qĎ Lppµ; ¨V q : f P L

ppµ; ¨V q

(

, (2.21)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property


›

›rf sLqpµ;¨V q

›

›

Lqpµ;¨V q“ fLqpµ;¨V q

. (2.22)

In the setting of Definition 2.2.12 we do in the following not distinguish betweenan element f P Lppµ; ¨V q of Lppµ; ¨V q and its equivalence class in Lppµ; ¨V q.


2.3 Simple functions

The idea of the Lebesgue integral for real valued functions is to approximate thefunction by suitable simpler functions and then to define the Lebesgue integral ofthe “complicated” function as the limit of the integrals of the simpler functions.

Definition 2.3.1 (Simple functions). Let pΩ1,F1q and pΩ2,F2q be measurable spacesand let f : Ω1 Ñ Ω2 be an F1/F2-measurable function with the property that the setfpΩ1q is finite. Then f is called F1/F2-simple.

2.4 Strongly measurable functions

2.4.1 Separability

(Unfortunately) Measurable functions can, in general, not be approximated pointwise(see (2.16) in Corollary 2.2.10) by simple functions; see Theorem 2.4.7 below fordetails. To overcome this difficulty, we introduce the notion of a strongly measurablefunction; see Definition 2.4.3 below. For this the next definition is needed.

Definition 2.4.1 (Separability). A topological space pE, Eq is called separable if thereexist an at most countable set A Ď E such that A “ E.

A topological space that is not separable is in a certain sense extremely large.This, in turn, can cause several serious difficulties in the analysis of such spaces. Anexample of a non-separable topological space can be found below. Let a, b P R witha ă b. Then the R-Banach space pCpra, bs,Rq, ¨Cpra,bs,Rqq of continuous functions

from the interval ra, bs to R equipped with the supremum norm ¨Cpra,bs,Rq is anexample of a separable topological space as the set

"

f P Cpr0, 1s,Rq :

ˆ

Dn P N0 : Dλ0, . . . , λn P Q : @x P r0, 1s : fpxq “nř

k“0

λkxk

˙*

(2.23)is a countable dense subset of Cpr0, 1s,Rq.

Lemma 2.4.2. Let pE, dEq be a separable metric space and let A Ď E. Then themetric space pA, dE|AˆAq is separable.

Proof of Lemma 2.4.2. W.l.o.g. we assume that A ‰ H. Let penqnPN Ď E be asequence of elements in E such that the set ten P E : n P Nu is dense in E. In the

2.4. STRONGLY MEASURABLE FUNCTIONS 33

next step let pfnqnPN Ď A be a sequence of elements in A such that for all n P N itholds that

dEpfn, enq ď

#

0 : en P A

distEpA, tenuq `1

2n: en R A

. (2.24)

Then the set tfn P A : n P Nu is dense in A. Indeed, if v P A X ten P E : n P Nu,then

distEptvu, tf1, f2, . . . uq “ 0 (2.25)

and if v P Azten P E : n P Nu, then it holds for all n P N that

distEptvu, tf1, f2, . . . uq ď distEptvu, tfn, fn`1, . . . uq

“ infmPtn,n`1,... u

dEpv, fmq

ď infmPtn,n`1,... u

rdEpv, emq ` dEpem, fmqs


„

dEpv, emq ` distEpA, temuq `1

2m


„

2 dEpv, emq `1

2m

ď 2

„

infmPtn,n`1,... u

dEpv, emq

`1

2n

“ 2 distEptvu, ten, en`1, . . . uq `1

2n“

1

2n.

(2.26)


2.4.2 Strongly measurable functions

Definition 2.4.3 (Strongly measurable functions). Let pΩ,Fq be a measurable spaceand let pE, dEq be a metric space. A function f : Ω Ñ E is called strongly F/pE, dEq-measurable (or simply: strongly measurable) if f : Ω Ñ E is F/BpEq-measurable andif pfpΩq, dE|fpΩqˆfpΩqq is separable.

Let pΩ,Fq be a measurable space and let pE, dEq be a separable metric space.Lemma 2.4.2 then shows that every F/BpEq-measurable mapping f : Ω Ñ E is alsostrongly F/pE, dEq-measurable.

Exercise 2.4.4. Give an example of a measurable space pΩ,Fq, of a metric spacepE, dEq, and of an F/BpEq-measurable function f : Ω Ñ E which is not stronglyF/pE, dEq-measurable. Show that f is F/BpEq-measurable but not strongly F/pE, dEq-measurable.


2.4.3 Pointwise approximations of strongly measurable func-tions

As mentioned above, measurable functions can, in general, not be approximatedpointwise by simple functions. However, strongly measurable functions can be ap-proximated pointwise by simple functions. This is the subject of the Theorem 2.4.7below (cf., e.g., Lemma 1.1 in Da Prato & Zabczyk [7] and Lemma A.1.4 in Prevot& Rockner [24]). In the proof of Theorem 2.4.7 the following two lemmas are used.

Lemma 2.4.5 (Projections in metric spaces). Let pE, dEq be a metric space, letn P N, let e1, . . . , en P E and let Ppe1,...,enq : E Ñ E be the function with the propertythat for all x P E it holds that

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dEpek,xq“distEptxu,te1,...,enuqu. (2.27)

Then Ppe1,...,enq is BpEq/PpEq-measurable and for all x P E it holds that

dEpx, Ppe1,...,enqpxqq “ distEptxu, te1, . . . , enuq. (2.28)

Proof of Lemma 2.4.5. Identity (2.28) is an immediate consequence of (2.27). LetD “ pD1, . . . , Dnq : E Ñ Rn be the function with the property that for all x P E itholds that

Dpxq “ pD1pxq, . . . , Dnpxqq “ pdEpx, e1q, . . . , dEpx, enqq . (2.29)

Observe that D is continuous and hence that D is BpEq/BpRnq-measurable. Thisimplies that for all k P t1, 2, . . . , nu it holds that

P´1pe1,...,enq

pekq “

x P E : Ppe1,...,enqpxq “ ek(

“

!

x P E : k “ min

l P t1, 2, . . . , nu : dEpel, xq “ distptxu, te1, . . . , enuq(

)

“

"

x P E : k “ min

"

l P t1, 2, . . . , nu : Dlpxq “ minuPt1,...,nu

Dupxq

**

“

"

x P E :

ˆ

Dkpxq ď minuPt1,...,nu

Dupxq and Dkpxq ă minuPt1,...,k´1u

Dupxq

˙*

“

"

x P E :

ˆ

@ l P t1, . . . , k ´ 1u : Dkpxq ă Dlpxq and@ l P t1, . . . , nu : Dkpxq ď Dlpxq

˙*

“

»

—

–

k´1č

l“1

tx P E : Dkpxq ă Dlpxquloooooooooooooomoooooooooooooon

PBpEq

fi

ffi

fl

č

»

—

–

nč

l“1

tx P E : Dkpxq ď Dlpxquloooooooooooooomoooooooooooooon

PBpEq

fi

ffi

fl

P BpEq.

(2.30)


This, in turn, implies that for all A P PpEq it holds that

P´1pe1,...,enq

pAq “ P´1pe1,...,enq

`

AX te1, . . . , enu˘

“ YfPAXte1,...,enu P´1pe1,...,enq

pfqlooooomooooon

PBpEq

P BpEq. (2.31)


Lemma 2.4.6. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space andlet f : Ω Ñ E be a function, and let gn : Ω Ñ E, n P N, be a sequence of stronglyF/pE, dEq-measurable functions such that for all ω P Ω it holds that

limnÑ8


Then f is strongly F/pE, dEq-measurable.

Proof of Lemma 2.4.6. Corollary 2.2.10 ensures that f is F/BpEq-measurable. Itthus remains to prove that fpΩq is separable. This follows from Lemma 2.4.2 andfrom the fact that YnPNgnpΩq is separable. The proof of Lemma 2.4.6 is thus com-pleted.

We now present the promised pointwise approximation result for strongly mea-surable functions.

Theorem 2.4.7 (Approximations of strongly measurable functions). Let pΩ,Fq bea measurable space, let pE, dEq be a metric space, and let f : Ω Ñ E be a function.Then the following four statements are equivalent:

(i) It holds that f is F/pE, dEq-strongly measurable.

(ii) There exists a sequence gn : Ω Ñ E, n P N, of strongly F/pE, dEq-measurablefunctions with the property that for all ω P Ω it holds that

limnÑ8


(iii) There exists a sequence gn : Ω Ñ E, n P N, of simple functions with the prop-erty that for all ω P Ω it holds that

limnÑ8


(iv) There exists a sequence gn : Ω Ñ E, n P N, of simple functions with the prop-erty that for all ω P Ω it holds that dEpfpωq, gnpωqq P r0,8q, n P N, decreasesmonotonically to zero.


Proof of Theorem 2.4.7. Clearly, (iv) implies (iii) and (iii) implies (ii). Lemma 2.4.6shows that (ii) implies (i). It thus remains to prove that (i) implies (iv). For this letf : Ω Ñ E be a strongly F/pE, dEq-measurable function. The fact that f is stronglyF/pE, dEq-measurable ensures that fpΩq is separable. Hence, there exists a sequencepenqnPN Ď fpΩq of elements in fpΩq with the property that

ten P fpΩq : n P Nu Ě fpΩq. (2.35)

In the next step let Ppe1,...,enq : E Ñ E, n P N, and gn : Ω Ñ E, n P N, be thefunctions with the property that for all x P E, n P N it holds that

Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dEpek,xq“distEptxu,te1,...,enuqu (2.36)

andgn “ Ppe1,...,enq ˝ f. (2.37)

Lemma 2.4.5 and the fact that f is F/BpEq-measurable implies that for all n P N itholds that gn is F/BpEq-measurable. In addition, by definition it holds for all n P Nthat gnpΩq Ď te1, . . . , enu is a finite set. We hence get that for all n P N it holdthat gn is an F/BpEq-simple function. Moreover, note that (2.28) in Lemma 2.4.5ensures that for all ω P Ω, n P N it holds that

dEpfpωq, gnpωqq “ dE`

fpωq, Ppe1,...,enqpfpωqq˘

“ distEpfpωq, te1, . . . , enuq . (2.38)

This and the property that for all ω P Ω it holds that distEpfpωq, te1, e2, . . . uq “ 0imply that for all ω P Ω, n P N it holds that

dEpfpωq, gnpωqq ě dEpfpωq, gn`1pωqq and limnÑ8



2.4.4 Sums of strongly measurable functions

The next result, Corollary 2.4.8, shows that the sum of two strongly measurable map-pings is again a strongly measurable mapping. Corollary 2.4.8 follows immediatelyfrom Theorem 2.4.7.

Corollary 2.4.8. Let pΩ,Fq be a measurable space, let pV, ¨V q be a normed vectorspace and let f, g : Ω Ñ V be strongly F/pV, ¨V q-measurable mappings. Then f `g : Ω Ñ V is strongly F/pV, ¨V q-measurable.


2.4.5 Lp-spaces of strongly measurable functions for p P r0,8q

Definition 2.4.9 (Lp-spaces for p P r0,8q). Let K P tR,Cu, q P p0,8q, let pΩ,A, µqbe a measure space, and let pV, ¨V q be a normed K-vector space. Then we denoteby L0pµ; ¨V q the set given by

L0pµ; ¨V q “ tf PMpΩ, V q : f is strongly A/pV, ¨V q-measurableu, (2.40)

we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that

for all f P L0pµ; ¨V q it holds that

fLqpµ;¨V q“

„ż

Ω

fpωqqV µpdωq

1q

P r0,8s, (2.41)

and we denote by Lqpµ; ¨V q the set given by

Lqpµ; ¨V q “

f P L0pµ; ¨V q : fLqpµ;¨V q

ă 8(

. (2.42)

Corollary 2.4.8 proves, in the setting of Definition 2.4.9, that for all p P r0,8q itholds that Lppµ; ¨V q is a K-vector space.

Definition 2.4.10 (Lp-spaces for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Thenwe denote by r¨sL0pµ;¨V q

: L0pµ; ¨V q Ñ PpL0pµ; ¨V qq the function with the property


rf sL0pµ;¨V q“

!

g P L0pµ; ¨V q : f ´ gL1pµ;¨V q

“ 0)

, (2.43)

we denote by Lppµ; ¨V q the set given by

Lppµ; ¨V q “

rf sL0pµ;¨V qĎ L0

pµ; ¨V q : f P Lppµ; ¨V q

(

, (2.44)

and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property

that for all f P L0pµ; ¨V q it holds that›

›rf sLqpµ;¨V q

›

›

Lqpµ;¨V q“ fLqpµ;¨V q

. (2.45)

In the setting of Definition 2.4.10 we do in the following not distinguish betweenan element f P Lppµ; ¨V q of Lppµ; ¨V q and its equivalence class in Lppµ; ¨V q.

Lemma 2.4.11. Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a measure space andlet pV, ¨V q be a K-Banach space. Then Lppµ; ¨V q is a K-Banach space too.


Lemma 2.4.12. Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a finite measure space,and let pV, ¨V q be a normed K-vector space. Then it holds that

rf sLppµ;¨V q: f is an F/BpV q-simple function

(

(2.46)

is dense in Lppµ; ¨V q.

Proof of Lemma 2.4.12. Let f P Lppµ; ¨V q. Theorem 2.4.7 proves that there existsa sequence gn : Ω Ñ V , n P N, of F/BpV q-simple functions with the property thatfor all ω P Ω it holds that fpωq ´ gnpωqV , n P N, decreases monotonically to zero.Lebesgue’s theorem of dominated convergence hence proves that

limnÑ8

f ´ gnLppµ;¨V q“ lim

nÑ8

„ż

Ω

fpωq ´ gnpωqpV µpdωq

1p

“ 0. (2.47)


2.5 Continuous functions

2.5.1 Topological spaces

Definition 2.5.1 (Topology). Let E be a set and let E Ď PpEq be a set such thatH, E P E, such that for all A Ď E it holds that YAPAA P E, and such that for allA,B P E it holds that pAXBq P E. Then E is called a topology on E and pE, Eq iscalled a topological space.

Proposition 2.5.2 (Topology induced by a function). Let E be a set, let T Ď R bea set, and let d : E ˆ E Ñ T be a function. Then it holds that the set

"

A P PpEq :´

@ v P A :“

D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰

¯

*

(2.48)

is a topology on E.

Proof of Proposition 2.5.2. Throughout this proof let E Ď PpEq be the set given by

E “"

A P PpEq :´

@ v P A :“


¯

*

. (2.49)

First, we observe thatH, E P E . Next we note that for all A Ď E and all v P rYAPAAsthere exists a set A P A and a real number ε P p0,8q such that v P A and such that

2.5. CONTINUOUS FUNCTIONS 39

tu P E : dpv, uq ă εu Ď A. In particular, this implies that for all A Ď E and allv P rYAPAAs there exists a real number ε P p0,8q such that tu P E : dpv, uq ă εu ĎrYAPAAs. Hence, we obtain that for all A Ď E it holds that

rYAPAAs P E . (2.50)

In the next step we observe that for all A,B P E and all v P pAXBq there existsreal numbers εA, εB P p0,8q such that

tu P E : dpv, uq ă εAu Ď A and tu P E : dpv, uq ă εBu Ď B. (2.51)

Hence, we obtain that for all A,B P E and all v P pAXBq there exists real numbersεA, εB P p0,8q such that

u P E : dpv, uq ă mintεA, εBu(

Ď pAXBq . (2.52)

This proves that for all A,B P E it holds that pAXBq P E . The proof of Proposi-tion 2.5.2 is thus completed.

Proposition 2.5.2 above ensures that the designation in the next definition isreasonable.

Definition 2.5.3 (Topology induced by a function). Let E be a set, let T Ď R be aset, and let d : E ˆ E Ñ T be a function. Then we denote by τpdq Ď PpEq the setgiven by

τpdq “

"

A P PpEq :´

@ v P A :“


¯

*

(2.53)

and we call τpdq the topology induced by d.

Lemma 2.5.4 (Balls are open). Let E be a set, let T Ď R be a set, let d : EˆE Ñ Tbe a function with the property that @x, y, z P E : dpx, zq ď dpx, yq ` dpy, zq, and letε P p0,8q, v P E. Then it holds that

tu P E : dpv, uq ă εu P τpdq. (2.54)

Proof of Lemma 2.5.4. First, observe that for all u P E with dpv, uq ă ε and allw P E with dpu,wq ă 1

2rε´ dpv, uqs it holds that

dpv, wq ď dpv, uq ` dpu,wq ă dpv, uq ` 12rε´ dpv, uqs “ 1

2dpv, uq ` ε

2ă ε. (2.55)


This implies that for all x P tu P E : dpv, uq ă εu there exists a real number δ P p0,8qsuch that

tu P E : dpx, uq ă δu Ď tu P E : dpv, uq ă εu . (2.56)

Hence, we obtain that tu P E : dpv, uq ă εu P τpdq. The proof of Lemma 2.5.4 is thuscompleted.

Proposition 2.5.5 (Convergence in the induced topology). Let E be a set, let d : EˆE Ñ r0,8q be a function with the property that @x P E : dpx, xq “ 0 and @x, y, z PE : dpx, zq ď dpx, yq ` dpy, zq, and let e : N0 Ñ E be a function. Then it holds thatlim supnÑ8 d

`

ep0q, epnq˘

“ 0 if and only if for all A P τpdq with ep0q P A there existsa natural number N P N such that for all n P tN,N ` 1, . . . u it holds that epnq P A.

Proof of Proposition 2.5.5. First of all, recall that lim supnÑ8 d`

ep0q, epnq˘

“ 0 ifand only if @ ε P p0,8q : DN P N : @n P tN,N ` 1, . . . u : d

`

ep0q, epnq˘

ă ε. Hence,we obtain that lim supnÑ8 d

`

ep0q, epnq˘

“ 0 if and only if for all ε P p0,8q thereexists a natural number N P N such that for all n P tN,N ` 1, . . . u it holds thatepnq P tu P E : dpep0q, uq ă εu. This and Lemma 2.5.4 complete the proof of Propo-sition 2.5.5.

2.5.2 Semi-metric spaces

Definition 2.5.6. Let E be a set and let d : E ˆ E Ñ r0,8q be a function with theproperty that for all x, y, z P E it holds that

(i) dpx, xq “ 0,

(ii) dpx, yq “ dpy, xq, and

(iii) dpx, zq ď dpx, yq ` dpy, zq.

Then we call d a semi-metric (on E) and we call pE, dq a semi-metric space.

Definition 2.5.7 (Globally bounded sets). Let pE, dq be a semi-metric space andlet A Ď E be a set with the property that @ e P E : supaPA dpa, eq ă 8. Then we saythat A is d-globally bounded (that A is globally bounded).

Lemma 2.5.8 (Globally bounded sets). Let pE, dq be a semi-metric space with E ‰H and let A Ď E be a set. Then A is d-globally bounded if and only if D e PE : supaPA dpa, eq ă 8.


Definition 2.5.9 (Globally bounded functions). Let E be a set, let pF, dq be a semi-metric space, and let f : E Ñ F be a function with the property that fpEq is ad-globally bounded set. Then we say that f is d-globally bounded (that f is globallybounded).

2.5.3 Continuity properties of functions

Definition 2.5.10 (Continuous functions). Let pE1, E1q and pE2, E2q be topologicalspaces. Then we denote by CpE1, E2q the set of all continuous functions from E1 toE2.

Definition 2.5.11 (Uniformly continuous). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function with the property that

@ ε P p0,8q : D δ P p0,8q : @x, y P E : ppdEpx, yq ă δq ñ pdF pfpxq, fpyqq ă εqq .(2.57)

Then we say that f is uniformly continuous (dE/dF -uniformly continuous).

Definition 2.5.12 (Holder continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces and let r P p0,8q. Then we denote by |¨|C0,rpE,F q : MpE,F q Ñ r0,8s

the function with the property that for all f PMpE,F q it holds that

|f |C0,rpE,F q “ sup

ˆ

t0u Y

"

dF pfpxq, fpyqq

|dEpx, yq|r P p0,8s : px, y P E, dF pfpxq, fpyqq ą 0q

*˙

(2.58)and we denote by C0,rpE,F q the set given by

C0,rpE,F q “

!

f PMpE,F q : |f |C0,rpE,F q ă 8

)

. (2.59)

Definition 2.5.13 (Holder continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces, let r P p0,8q, and let f P C0,rpE,F q. Then f is called an r-Holdercontinuous function (with respect to dE/dF )

Lemma 2.5.14. Let pV, ¨V q and pW, ¨W q be normed R-vector spaces, let U Ď V bean open set, let v P U , and let f : U Ñ W be a function which is Frechet differentiablein v. Then

lim suphŒ0

„

F pv ` hq ´ F pvqVhV

“ limεŒ0

suphPV zt0u,hV ďε

„

F pv ` hq ´ F pvqVhV

“ F 1pvqLpV,W q .

(2.60)


2.5.4 Modulus of continuity

Definition 2.5.15 (Modulus of continuity). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function. Then we denote by wdE ,dFf : r0,8s Ñ r0,8sthe function with the property that for all h P r0,8s it holds that

wdE ,dFf phq “ sup´

t0u Y!

dF`

fpxq, fpyq˘

P r0,8q :“

x, y P E with dEpx, yq ď h‰

)¯

(2.61)and we call wdE ,dFf the modulus of continuity of f .

Lemma 2.5.16 (Properties of the modulus of continuity). Let pE, dEq and pF, dF qbe semi-metric spaces and let f : E Ñ F be a function. Then

• wdE ,dFf is non-decreasing,

• f is dE/dF -uniformly continuous if and only if limhŒ0wdE ,dFf phq “ 0,

• f is dF -globally bounded if and only if wdE ,dFf p8q ă 8, and

• for all x, y P E it holds that dF`

fpxq, fpyq˘

ď wdE ,dFf

`

dEpx, yq˘

.

The proof of Lemma 2.5.16 is clear and therefore omitted.

Exercise 2.5.17. Give an example of a metric space pE, dq such that for all h Pr0,8s it holds that

wd,didEphq “

#

0 : h P r0, 1q

1 : h P r1,8s. (2.62)

Prove that your metric space has the desired properties.

2.5.5 Extensions of uniformly continuous functions

Lemma 2.5.18 (Uniformly continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces, let f : E Ñ F be a uniformly continuous function, and let penqnPN Ď Ebe a Cauchy sequence. Then fpenq P F , n P N, is a Cauchy sequence too.

Proof of Lemma 2.5.18. The assumption that penqnPN is a Cauchy sequence and theassumption that f is uniformly continuous imply that

limNÑ8

supn,mPtN,N`1,... u

dF pfpenq, fpemqq ď limNÑ8

supn,mPtN,N`1,... u

wdE ,dFf pdEpen, emqq

ď limNÑ8

wdE ,dFf

`

supn,mPtN,N`1,... u dEpen, emq˘

“ 0.(2.63)


This shows that fpenq P F , n P N, is a Cauchy sequence. The proof of Lemma 2.5.18is thus completed.

Proposition 2.5.19 (Extension of uniformly continuous functions). Let pE, dEq bea semi-metric space, let pF, dF q be a complete semi-metric space, let A Ď E, andlet f : A Ñ F be a uniformly continuous function. Then there exists a unique f PCpA, F q with the property that f |A “ f , it holds for all h P r0,8s that

wdE ,dFf phq ď wdE ,dFf

phq ď limεŒ0

wdE ,dFf ph` εq (2.64)

and it holds that f is uniformly continuous.

Proof of Proposition 2.5.19. The uniqueness of f is clear. It remains to prove theexistence of a function f with the desired properties. For this observe that for allx P A and all penqnPN Ď A, penqnPN Ď A with limnÑ8 en “ x “ limnÑ8 en it holdsthat

lim supnÑ8

d`

fpenq, fpenq˘

ď lim supnÑ8

wdE ,dFf

`

dpen, enq˘

“ 0. (2.65)

This, Lemma 2.5.18, and the assumption that pF, dF q is complete imply that thereexist a function f : AÑ F with the property that for all x P A and all penqnPN Ď Awith limnÑ8 en “ x it holds that

fpxq “ limnÑ8

fpenq. (2.66)

In the next step we note that the continuity of f implies that for all x P A it holdsthat fpxq “ fpxq. In the next step we show (2.64). The first inequality in (2.64)is clear. To prove the second inequality in (2.64) let h P r0,8s and let x0, y0 P Awith dEpx0, y0q ď h. Then there exist sequences pxnqnPN Ď A and pynqnPN Ď A withthe property that limnÑ8 xn “ x0 and limnÑ8 yn “ y0. This implies that for allε P p0,8q it holds that

dF`

fpx0q, fpy0q˘

“ dF

´

limnÑ8

fpxnq, limnÑ8

fpynq¯

“ limnÑ8

dF pfpxnq, fpynqq

“ lim infnÑ8

dF pfpxnq, fpynqq ď lim infnÑ8

wf`

dEpxn, ynq˘

ď wf`

dEpx0, y0q ` ε˘

.(2.67)

This proves the second inequality in (2.64). The second inequality in (2.64), inturn, shows that f is uniformly continuous. The proof of Proposition 2.5.19 is thuscompleted.


Exercise 2.5.20. Specify a metric space pE, dEq, a complete metric space pF, dF q,a set A Ď E, and a uniformly continuous function f : A Ñ F such that the uniquefunction f P CpA, F q with f |A “ f satisfies wf ‰ wf (i.e., there exists an h P r0,8ssuch that wf phq ‰ wf phq).

Remark 2.5.21. Let pE, dEq be a semi-metric space, let pF, dF q be a complete semi-metric space, let A Ď E be a dense subset of E, and let f : A Ñ F be a uniformlycontinuous function. Proposition 2.5.19 then proves that there exists a unique f PCpE,F q with f |A “ f . In the following we often write, for simplicity of presentation,f instead of f .

Lemma 2.5.22. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be semi-normed K-vector spaces, and let A : V Ñ W be a linear mapping. Then A is continuous if andonly if A is uniformly continuous.


Chapter 3

Linear functions

3.1 Linear spaces

We first recall some notions regarding linear spaces (also known as vector spaces).

Definition 3.1.1 (Span in a vector space). Let K be a field, let V be a K-vectorspace, and let A Ď V . Then we denote by spanV pAq Ď V the set with the propertythat

spanV pAq “ t0u Y tλ1a1 ` . . .` λnan P V : n P N, a1, . . . , an P A, λ1, . . . , λn P Ku .(3.1)

Definition 3.1.2 (Generating system). Let V be a vector space and let A Ď V be aset with the property that spanV pAq “ V . Then A is called a generating system (inV ).

Definition 3.1.3 (Linearly independent). Let K be a field, let V be a K-vector space,and let A Ď V be a set with the property that for all n P N, λ1, . . . , λn P K and allpairwise different a1, . . . , an P A with λ1a1 ` . . .` λnan “ 0 it holds that

λ1 “ . . . “ λn “ 0. (3.2)

Then A is called linearly independent (in V ).

Definition 3.1.4 (Basis of a vector space). Let V be a vector space and let A Ď Vbe a linearly independent generating system in V . Then A is called a (Hamel) basisof V .

Theorem 3.1.5 (Every vector space has a basis1). Let V be a vector space. Thenthere exists a subset A Ď V such that A is a basis of V .

1if one believes in the axiom of choice at least

45

46 CHAPTER 3. LINEAR FUNCTIONS

3.2 An intermezzo on sums over possibly uncount-

able index sets

3.2.1 Fubini’s theorem in the case of non-sigma-finite mea-sure spaces

Definition 3.2.1 (Counting measure on a set). Let A be a set. Then we denote by#A : PpAq Ñ r0,8s the counting measure on A.

Example 3.2.2. It holds that

ż

r0,1s

ż

r0,1s

1txupyq#RpdyqBorelRpdxq “

ż 1

0

ÿ

yPr0,1s

1txupyq dx “

ż 1

0

#Rptxuq dx “ 1,

ż

r0,1s

ż

r0,1s

1txupyq BorelRpdxq#Rpdyq “ÿ

yPr0,1s

ż 1

0

1txupyq dx “ÿ

yPr0,1s

BorelRptyuq “ 0.

(3.3)

3.2.2 Nets

Definition 3.2.3 (Relation). Let A and B be arbitrary sets and let C Ď AˆB be asubset of AˆB. Then the triple pA,B,Cq is called a (binary) relation (on pA,Bq).

Definition 3.2.4. Let „ “ pA,B,Cq be a relation, let a P A, and let b P B. Thenwe write a „ b if and only if pa, bq P C.

Definition 3.2.5 (Function). Let pA,B,Cq be a relation with the property that forevery a P A there exists exactly one b P B such that pa, bq P C. Then pA,B,Cq iscalled a function.

Definition 3.2.6 (Preorder). Let X be a set and let ĺ be a relation on pX,Xq withthe property that

(i) @x P X : x ĺ x (Reflexivity) and

(ii) @x, y, z P X :`

px ĺ y and y ĺ zq ñ px ĺ zq˘

(Transitivity).

Then the pair pX,ĺq is called a preorder (on X).

3.2. AN INTERMEZZOON SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS47

Definition 3.2.7 (Directed set). Let pX,ĺq be a preorder with the property that

@x, y P X : D z P X : px ĺ z and y ĺ zq . (3.4)

Then pX,ĺq is called a directed set.

Definition 3.2.8 (Nets). Let pX,ĺq be a directed set, let pE, Eq be a topologicalspace, and let φ : X Ñ E be a function from X to E. Then φ is called a net (frompX,ĺq to pE, Eq).

Definition 3.2.9 (Convergence of a net). Let pX,ĺq be a directed set, let pE, Eq bea topological space, let e P E, and let φ : X Ñ E be a net with the property that forevery neighbourhood U Ď E of e there exists an f0 P X such that for all f P X withf0 ĺ f it holds that φpfq P U . Then φ is said to converge (with respect to pX,ĺq) toe, in symbols,

limfPpX,ĺq

φpfq “ e. (3.5)

3.2.2.1 Confinal sequences

See, e.g., the book of Heuser [Analysis I, Satz 44.7] for this section.

Definition 3.2.10 (Confinal sequence). Let pX,ĺq be a directed set and let xn P X,n P N, be sequence with the property that for all y P X there exists a natural numberN P N such that for all n P tN,N ` 1, . . . u it holds that

y ĺ xn. (3.6)

Then pxnqnPN is called confinal (in pX,ĺq).

Proposition 3.2.11 (Convergence of confinal sequences). Let pX,ĺq be a directedset, let pE, Eq be a topological space, let e P E, let φ : X Ñ E be a net which convergesto e, and let xn P X, n P N, be a confinal sequence. Then it holds that the sequenceφpxnq, n P N, converges to e.

Proof of Proposition 3.2.11. Let U P E be an open set with the property that e P U .The assumption that φ converges to e ensures that there exists an element y P Xwith the property that for all z P X with y ĺ z it holds that

φpzq P U. (3.7)


In the next step we note that the assumption that xn, n P N, is confinal implies thatthere exists a natural number N P N such that for all n P tN,N ` 1, . . . u it holdsthat

y ĺ xn. (3.8)

Combining (3.7) and (3.8) proves that for all n P tN,N ` 1, . . . u it holds that

φpxnq P U. (3.9)


Proposition 3.2.12 (Convergence of nets). Let pX,ĺq be a directed set, let xn P X,n P N, be a confinal sequence, let pE, Eq be a topological space, let φ : X Ñ E be a net,and assume that for every confinal seqence yn P X, n P N, it holds that φpynq P E,n P N, is a convergent sequence. Then there exists an element e P E such that φconverges to e and such that for every confinal seqence yn P X, n P N, it holds thatφpynq P E, n P N, converges to e.

Proof of Proposition 3.2.12. Throughout this proof we denote by p¨qb p¨q : MpN, Xq2 ÑMpN, Xq the mapping with the property that for all y, z : NÑ X it holds that

py b zqn “

#

zn2 : n is even

ypn`1q2 : n is odd(3.10)

Next observe that for all confinal sequences y, z : N Ñ X it holds that y b z is aconfinal sequence. By assumption we hence obtain that for all confinal sequencesy, z : NÑ X it holds that

limnÑ8

φ pynq “ limnÑ8

φ`

py b zqn˘

“ limnÑ8

φ pznq . (3.11)

This proves that there exists an element e P E such that for every confinal sequencey : NÑ X it holds that

limnÑ8

φpynq “ e. (3.12)

We now complete the proof of Proposition 3.2.12 by a contradiction, that is, weassume that φ does not converge to e. Hence, there exists an open set U P E withthe property that e P U and with the property that for all y P X there exists anelement z P X such that

y ĺ z and φpzq R U. (3.13)

3.2. AN INTERMEZZOON SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS49

This proves, in particular, that there exists a function z : N Ñ X such that for alln P N it holds that

xn ĺ zn and φpznq R U. (3.14)

The assumption that xn P X, n P N, is confinal and the fact that @n P N : xn ĺ znproves that z : N Ñ X is confinal too. This and (3.12) contradict to (3.14). Theproof of Proposition 3.2.12 is thus completed.

Example 3.2.13 (cf. Exercise 6 a) in Heuser). Let a, b P R with a ă b, let X be theset given by

X “ tA P Ppra, bsq : #RpAq ă 8u , (3.15)

and let ĺ be the relation on X with the property that for all A,B P X it holds thatA ĺ B if and only if A Ď B. Then there exists no sequence x : N Ñ X which isconfinal in pX,ĺq. Indeed, observe that for every sequence x : N Ñ X and everyt P

`

ra, bszpYnPNxnq˘

there exists no N P N such that ttu ĺ xn.

3.2.3 Sums over possibly uncountable index sets

Definition 3.2.14. Let A be a set and let f : A Ñ r0,8s be a function. Then wedenote by

ř

aPA fpaq the extended real number in r0,8s with the property that

ÿ

aPA

fpaq “

ż

A

fpaq#Apdaq. (3.16)

Another way to define the sum in (3.16) above is to employ the concept of a net.This is illustrated in the following example.

Example 3.2.15 (Sums through nets). Let A be a set, let f : AÑ r0,8s be a func-tion, and let φ : tx P PpAq : #Apxq ă 8u Ñ r0,8s be the function with the propertythat for all finite subsets x Ď A of A it holds that

φpxq “ÿ

aPx

fpaq. (3.17)

Then it holds that the pair

ptx P PpAq : #Apxq ă 8u ,Ďq (3.18)

is a directed set and it holds that φ is a net which converges toř

aPA fpaq.


3.2.4 Fubini for sums

Lemma 3.2.16. Let A be a set and let f : AÑ r0,8s be a function withř

aPA fpaq ă8. Then it holds that the set f´1pp0,8sq is at most countable.

Proof of Lemma 3.2.16. Monotonicity proves that for all ε P p0,8q it holds that

#A

`

f´1prε,8qq

˘

¨ ε ďÿ

aPA

fpaq ă 8. (3.19)

This shows that for all n P N it holds that the set f´1`

r1n,8q˘

is finite. This impliesthat the set

f´1`

p0,8q˘

“ YnPNf´1`

r1n,8q˘

(3.20)

is at most countable. The proof of Lemma 3.2.16 is thus completed.

Lemma 3.2.17 (Fubini for sums). Let A and B be sets and let f : A ˆ B Ñ r0,8sbe a function. Then

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

bPB

ÿ

aPA

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq. (3.21)

Proof of Lemma 3.2.17. W.l.o.g. we assume that pA ˆ Bq ‰ H is a non-empty set.We prove that

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq. (3.22)

Clearly, (3.22) implies (3.21). To prove (3.22), we distinguish between several cases.In the first case we assume that

ÿ

pa,bqPAˆB

fpa, bq ă 8. (3.23)

This assumption together with Lemma 3.2.16 implies that there exists a sequencepan, bnq P Aˆ B, n P N, such that for all px, yq P pAˆ Bqztpan, bnq : n P Nu it holdsthat fpx, yq “ 0. The theorem of Fubini hence proves that

ÿ

pa,bqPAˆB

fpa, bq “ÿ

pa,bqPtan : nPNuˆtbn : nPNu

fpa, bq

“

ż

fpa, bq #tan : nPNuˆtbn : nPNupda, dbq

“

ż ż

fpa, bq #tbn : nPNupdbq #tan : nPNupdaq

“ÿ

aPtan : nPNu

ÿ

bPtbn : nPNu

fpa, bq “ÿ

aPA

ÿ

bPB

fpa, bq.

(3.24)

3.3. HILBERT SPACES 51

This finishes the proof of (3.22) in the caseÿ

pa,bqPAˆB

fpa, bq ă 8. (3.25)

In the second case we assume thatÿ

aPA

ÿ

bPB

fpa, bq ă 8. (3.26)

Lemma 3.2.16 implies then that there exists an at most countable set A Ď A suchthat for all a P AzA it holds that

ř

bPB fpa, bq “ 0. Moreover, again Lemma 3.2.16implies that there exist at most countable sets Ba Ď B, a P A, such that for alla P A, b P BzBa it holds that fpa, bq “ 0. The theorem of Fubini hence shows that

ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

aPA

ÿ

bPB

fpa, bq “ÿ

aPA

ÿ

bPBa

fpa, bq “ÿ

aPA

ÿ

bPpYaPABaq

fpa, bq

“ÿ

pa,bqPAˆpYaPABaq

fpa, bq “ÿ

pa,bqPAˆB

fpa, bq.(3.27)


3.3 Hilbert spaces

In the next step we recall some notions regarding Hilbert spaces.

Definition 3.3.1 (Orthogonal). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H. Then A is called orthogonal (in pH, 〈¨, ¨〉H , ¨Hq/in H) if for all a, b P Awith a ‰ b it holds that

〈a, b〉H “ 0. (3.28)

Definition 3.3.2 (Orthonormal). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H be orthogonal. Then A is called orthonormal (in pH, 〈¨, ¨〉H , ¨Hq/in H) iffor all a P A it holds that

aH “ 1. (3.29)

Definition 3.3.3 (Completeness). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space. A setA Ď H is called complete (in pH, 〈¨, ¨〉H , ¨Hq/in H) if spanHpAq is dense in H, i.e.,if

spanHpAq “ H. (3.30)


Definition 3.3.4 (Orthonormal basis). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space andlet A Ď H be a complete and orthonormal set in H. Then A is called an orthonormalbasis of pH, 〈¨, ¨〉H , ¨Hq/of H.

Question 3.3.5. Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and let A Ď H be anorthonormal basis of H. Is A then a basis of H?

Theorem 3.3.6 (Every Hilbert space has an orthonormal basis2). Let pH, 〈¨, ¨〉H , ¨Hqbe a Hilbert space. Then there exists an orthonormal basis A Ď H of H.

Proposition 3.3.7 (A characterization for separable Hilbert spaces). LetK P tR,Cuand let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space. Then H is separable if and only if thereexists an at most countable orthonormal basis A Ď H of H.

Proof of Proposition 3.3.7. W.l.o.g. we assume that H ‰ t0u. If A Ď H is an atmost countable orthonormal basis of H, then the set

tλ1a1 ` . . .` λnan : n P N, a1, . . . , an P A, λ1, . . . , λn P tx P K : Repxq, Impxq P Quu(3.31)

is an at most countable dense subset of H. This proves the “ð” direction in thestatement of Proposition 3.3.7. The “ñ” direction in in the statement of Proposi-tion 3.3.7 follows from an application of the Gram-Schmidt process.

Example 3.3.8. Let er : R Ñ R, r P R, be the functions with the property that forall r, x P R it holds that

erpxq “

#

1 : x “ r

0 : x ‰ r. (3.32)

Then note that ter P L2p#R; |¨|

Rq : r P Ru is an orthonormal basis of the Hilbert space

L2p#R; |¨|Rq. Proposition 3.3.7 hence proves that the Hilbert space L2p#R; |¨|

Rq is not

separable.

Lemma 3.3.9. Let n P N, x “ px1, . . . , xnq P Rn, p P r1,8q. Then

xLpp#t1,2,...,nu;|¨|Rq“ p|x1|

p` . . .` |xn|

pq1pď |x1| ` . . .` |xn| “ xL1p#t1,2,...,nu;|¨|Rq

.

(3.33)

2if one believes in the axiom of choice at least


Proof of Lemma 3.3.9. Throughout this proof let e1, . . . , en P Rn be the vectors given

by e1 “ p1, 0, . . . , 0q, e2 “ p0, 1, 0, . . . , 0q, . . . , en “ p0, . . . , 0, 1q. Next observe thatthe triangle inequality implies that

xLpp#t1,2,...,nu;|¨|Rq“

›

›

›

›

›

nÿ

k“1

xkek

›

›

›

›

›

Lpp#t1,2,...,nu;|¨|Rq

ď

nÿ

k“1

xkekLpp#t1,2,...,nu;|¨|Rq

“

nÿ

k“1

|xk| ekLpp#t1,2,...,nu;|¨|Rq“

nÿ

k“1

|xk| “ xL1p#t1,2,...,nu;|¨|Rq.

(3.34)


3.3.1 Best approximations and projections in Hilbert spaces

Theorem 3.3.10 (Best approximation in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be aHilbert space, let U Ď H be a closed subspace of H, and let v P H. Then there existsa unique u P U with the property that

u´ vH “ infwPU

w ´ vH . (3.35)

Theorem 3.3.11 (Projection in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbertspace and let U Ď H be a closed subspace of H. Then there exists a unique P P LpHqwith the property that P pHq Ď U and with the property that for all v P H it holdsthat

P pvq ´ vH “ infwPU

w ´ vH . (3.36)

Definition 3.3.12 (Projection in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbertspace and let U Ď H be a closed subspace of H. Then we denote by PU P LpHq theunique bounded linear operator from H to H with the property that PUpHq Ď U andwith the property that for all v P H it holds that

PUpvq ´ vH “ infwPU

w ´ vH (3.37)

and we call PU the projection of H on U .


3.3.2 Examples of orthonormal bases

3.3.2.1 Trigonometric functions

Lemma 3.3.13 (Real and imaginary part of product of complex numbers). For allz1, z2 P C it holds that

Repz1 ¨ z2q “ Repz1q ¨Repz2q ´ Impz1q ¨ Impz2q , (3.38)

Impz1 ¨ z2q “ Repz1q ¨ Impz2q ` Impz1q ¨Repz2q . (3.39)

The proof of Lemma 3.3.13 is clear. The next lemma presents a well-knownidentity for the difference of two arguments of the cosine function.

Lemma 3.3.14. For all x, y P R it holds that

cosp2xq ´ cosp2yq “ 2 sinpy ´ xq sinpy ` xq. (3.40)

Proof of Lemma 3.3.14. Throughout this proof let ϕ, ϕ0,1, ϕ1,1 P C8pR2,Rq be the

functions with the property that for all x, y P R it holds that

ϕpx, yq “ cosp2xq ´ cosp2yq ´ 2 sinpy ´ xq sinpy ` xq,

ϕ0,1px, yq “`

B

Byϕ˘

px, yq, ϕ1,1px, yq “`

B2

BxByϕ˘

px, yq.(3.41)

Next observe that for all x, y P R it holds that

ϕpx, yq “ ϕpx, 0q `

ż y

0

ϕ0,1px, sq ds

“ ϕpx, 0q `

ż y

0

„

ϕ0,1p0, sq `

ż x

0

ϕ1,1pr, sq dr

ds

“ ϕpx, 0q `

ż y

0

ϕ0,1p0, sq ds`

ż y

0

ż x

0

ϕ1,1pr, sq dr ds.

(3.42)

Moreover, observe that for all x, y P R it holds that

ϕ0,1px, yq “ 2 sinp2yq ´ 2 cospy ´ xq sinpy ` xq ´ 2 sinpy ´ xq cospy ` xq, (3.43)

ϕ0,1p0, yq “ 2 sinp2yq ´ 4 sinpyq cospyq, (3.44)


ϕ1,1px, yq “ ´2 sinpy ´ xq sinpy ` xq ´ 2 cospy ´ xq cospy ` xq

` 2 cospy ´ xq cospy ` xq ` 2 sinpy ´ xq sinpy ` xq “ 0.(3.45)

Equation (3.44) and Lemma 3.3.13 imply that for all y P R it holds that

ϕ0,1p0, yq “ 2 psinpy ` yq ´ cospyq sinpyq ´ sinpyq cospyqq

“ 2`

Im`

eiy ¨ eiy˘

´Re`

eiy˘

¨ Im`

eiy˘

´ Im`

eiy˘

¨Re`

eiy˘˘

“ 0.(3.46)

This, (3.45) and (3.42) prove that for all x, y P R it holds that

ϕpx, yq “ ϕpx, 0q “ cosp2xq ´ 1´ 2 sinp´xq sinpxq “ cosp2xq ´ 1` 2 |sinpxq|2

“ cosp2xq ` |sinpxq|2 ´ |cospxq|2

“ Re`

eix ¨ eix˘

´“

Re`

eix˘

¨Re`

eix˘

´ Im`

eix˘

¨ Im`

eix˘‰

.

(3.47)

This and Lemma 3.3.13 imply that for all x, y P R it holds that ϕpx, yq “ 0. Theproof of Lemma 3.3.14 is thus completed.

3.3.2.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq

Proposition 3.3.15. The sets

p?

2 sinpnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

, (3.48)

p?

2 cospnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

Y

1(

, (3.49)

1(

Y

p?

2 sinp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

Y

p?

2 cosp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

,(3.50)

p?

2 sinppn´ 12qπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

(3.51)

are orthonormal in L2pBorelp0,1q; |¨|Rq.


Proof of Proposition 3.3.15. First of all, note that for all n P N it holds thatż 1

0

cospnπxq dx “ 0. (3.52)

Next observe that integration by parts proves that for all n,m P N it holds thatż 1

0

sinpnπxq sinpmπxq dx

“

„

´ cospnπxq sinpmπxq

nπ

x“1

x“0

`mπ

nπ

ż 1

0

cospnπxq cospmπxq dx

“mπ

nπ

ż 1

0

cospnπxq cospmπxq dx.

(3.53)

In addition, observe that (3.38) together with (3.52) ensures that for all n,m P N itholds that

ż 1

0

sinpnπxq sinpmπxq dx “

ż 1

0

Im`

einπx˘

¨ Im`

eimπx˘

dx

“

ż 1

0

Re`

einπx˘

¨Re`

eimπx˘

´Re`

einπx ¨ eimπx˘

dx

“

ż 1

0

cospnπxq cospmπxq dx´

ż 1

0

Re`

eipn`mqπx˘

dx

“

ż 1

0

cospnπxq cospmπxq dx.

(3.54)

Putting (3.54) into (3.53) proves that for all n,m P N with n ‰ m it holds thatż 1

0

sinpnπxq sinpmπxq dx “

ż 1

0

cospnπxq cospmπxq dx “ 0. (3.55)

In addition, observe that (3.54) implies that for all n P N it holds that

1 “

ż 1

0

|sinpnπxq|2 ` |cospnπxq|2looooooooooooooomooooooooooooooon

“1

dx “ 2

ż 1

0

|sinpnπxq|2 dx. (3.56)

This, (3.54) and (3.55) imply that for all n,m P N it holds thatż 1

0

?2 sinpnπxq ¨

?2 sinpmπxq dx “

ż 1

0

?2 cospnπxq ¨

?2 cospmπxq dx

“

#

0 : n ‰ m

1 : n “ m.

(3.57)


This and (3.56) prove that the set (3.48) is orthonormal in L2pBorelp0,1q; |¨|Rq. Fur-thermore, note that for all n P N it holds that

ż 1

0

1 ¨?

2 cospnπxq dx “?

2

ż 1

0

cospnπxq dx “?

2

„

sinpnπxq

nπ

x“1

x“0

“?

2

„

sinpnπq ´ sinp0q

nπ

“ 0.

(3.58)

This, (3.57) and (3.56) show that the set (3.49) is an orthonormal set in L2pBorelp0,1q; |¨|Rq.In the next step we observe that (3.38) and (3.52) imply that for all n,m P N it holdsthat

ż 1

0

sinppn´ 12qπxq sinppm´ 12qπxq dx

“

ż 1

0

Im`

eipn´12qπx˘

¨ Im`

eipm´12qπx˘

dx

“

ż 1

0

Re`

eipn´12qπx˘

¨Re`

eipm´12qπx˘

dx´Re`

eipn´12qπx¨ eipm´12qπx

˘

dx

“

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx´Re

ˆż 1

0

eipn`m´1qπx dx

˙

“

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx.

(3.59)

Furthermore, integration by parts proves that for all n,m P N it holds thatż 1

0

sinppn´ 12qπxq sinppm´ 12qπxq dx

“

„

´ cosppn´ 12qπxq sinppm´ 12qπxq

pn´ 12qπ

x“1

x“0

`pm´ 12qπ

pn´ 12qπ

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx

“pm´ 12qπ

pn´ 12qπ

ż 1

0

cosppn´ 12qπxq cosppm´ 12qπxq dx.

(3.60)

As above we combine (3.59) and (3.60) to obtain that the set (3.51) is orthonormalin L2pBorelp0,1q; |¨|Rq.

Definition 3.3.16 (Hausdorff space). Let pE, Eq be a topological space with the prop-erty that for all a, b P E with a ‰ b there exists A,B P E with a P A, b P B andAXB “ H. Then pE, Eq is called a Hausdorff space.


Theorem 3.3.17 (Stone-Weierstrass). Let K P tR,Cu, let pE, Eq be a compactHausdorff space, and let A Ď CpE,Kq be a subalgebra of CpE,Kq such that

(i) @ v P A : v P A (A is a sub-*-algebra of CpE,Kq),

(ii) 1 P A (A is a sub-*-algebra of CpE,Kq with 1) and

(iii) @x, y P E, x ‰ y : D v P A : vpxq ‰ vpyq (A seperates points).

Then A is dense in CpE,Kq.

A proof of Theorem 3.3.17 in German language can, for example, be found inHeuser [11]. We illustrate Theorem 3.3.17 by the following result.

Proposition 3.3.18. Let S “ tpa, bq P R2 : a2 ` b2 “ 1u Ď R2, let arg : S Ñ r0, 2πqbe the function with the property that for all x P S it holds that

`

cospargpxqq, sinpargpxqq˘

“ x, (3.61)

and let A Ď CpS,Cq be the set given by

A “ď

NPN

"ˆ

Nř

n“Ń

an ei n argpxq

˙

xPS

P CpS,Cq : aŃ , a1Ń , . . . , aN P C

*

. (3.62)

Then A is dense in CpS,Cq.

Proof of Proposition 3.3.18. We prove Proposition 3.3.18 through an application ofTheorem 3.3.17. For this we note that for all v P A it holds that v P A. Moreover,note that

`

ei 0 argpxq˘

xPS“`

e0˘

xPS“ 1 P A. (3.63)

Furthermore, observe that for all n,m P Z, x P S it holds that

ei n argpxq¨ eim argpxq

“ ei pn`mq argpxq (3.64)

This ensures that A is a subalgebra of CpS,Cq. In order to apply Theorem 3.3.17, itremains to verify that A separates points. For this observe that pei argpxqqxPS P CpS,Cqand that for all x, y P S with argpxq ą argpyq it holds that

ei argpxq

ei argpyq“ eipargpxqárgpyqq

‰ 1. (3.65)

This ensures that for all x, y P S with x ‰ y it holds that

ei argpxq‰ ei argpyq. (3.66)

We can thus apply Theorem 3.3.17 to obtain that A is dense in CpS,Cq. The proofof Proposition 3.3.18 is thus completed.


Corollary 3.3.19. It holds that the set!

`

1?2πeinx

˘

xPp0,2πqP L2

pBorelp0,2πq; |¨|Cq : n P Z)

(3.67)

is an orthonormal basis of L2pBorelp0,2πq; |¨|Cq.

Proof of Corollary 3.3.19. Observe that for all n,m P Z it holds that

ż 2π

0

1?2πeinx 1?

2πeimx dx “

1

2π

ż 2π

0

eipmńqx dx

“

$

&

%

”

12πipmńq

eipmńqxıx“2π

x“0“ 0 : m ‰ n

1 : m “ n.

(3.68)

This proves that the set `

1?2πeinx

˘

xPp0,2πqP L2pBorelp0,2πq; |¨|Cq : n P Z

(

is orthonor-

mal in L2pBorelp0,2πq; |¨|Cq. Next we denote by

CP pr0, 2πs,Cq “ tf P Cpr0, 2πs,Cq : fp0q “ fp2πqu (3.69)

the set of all 2π-periodic continuous functions from r0, 2πs to C. Proposition 3.3.18implies that

span!

`

1?2πeinx

˘

xPp0,2πq: n P Z

)CP pr0,2πs,Cq

“ CP pr0, 2πs,Cq. (3.70)

This implies that

span!

`

1?2πeinx

˘

xPp0,2πq: n P Z

)L2pBorelp0,2πq;|¨|Cq

Ě CP pr0, 2πs,CqL2pBorelp0,2πq;|¨|Cq

“ L2pBorelp0,2πq; |¨|Cq.

(3.71)


Exercise 3.3.20. Prove that the sets!

pcospnxqqxPp0,πq P L2pBorelp0,πq; |¨|Rq : n P N

)

Y t1u (3.72)

and!

psinpnxqqxPp0,πq P L2pBorelp0,πq; |¨|Rq : n P N

)

(3.73)

are orthonormal bases of L2pBorelp0,πq; |¨|Rq.


Proposition 3.3.21. The sets

p?

2 sinpnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

, (3.74)

p?

2 cospnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

Y

1(

, (3.75)

1(

Y

p?

2 sinp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

Y

p?

2 cosp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

,(3.76)

p?

2 sinppn´ 12qπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N

(

(3.77)

are orthonormal bases of L2pBorelp0,1q; |¨|Rq.

Proposition 3.3.22 (Haar functions). Let Hn,k : p0, 1q Ñ R, k P t1, 2, . . . , 2nu,n P N0, be the functions with the property that for all n P N0, k P t1, 2, . . . , 2nu,t P p0, 1q it holds that

Hn,kptq “”

1`k´12n

,k´12

2n

˘ptq ´ 1`k´12

2n,k

2n

˘ptqı

2n2. (3.78)

Then it holds that the set

1 P L2pBorelp0,1q; |¨|Rq

(

Y

Hn,k P L2pBorelp0,1q; |¨|Rq : k P t1, 2, . . . , 2

nu, n P N0

(

(3.79)is an orthonormal basis of pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq.

Proof of Proposition 3.3.22. First of all, we observe that for all n1, n2 P N0, k1 P

t1, 2, . . . , 2n1u, k2 P t1, 2, . . . , 2n2u it holds that

ż 1

0

Hn1,k1ptq ¨Hn2,k2ptq dt “

#

1 : pn1, k1q “ pn2, k2q

0 : pn1, k1q ‰ pn2, k2q(3.80)

andż 1

0

Hn1,k1ptq dt “ 0. (3.81)

This proves that the set


(

Y


nu, n P N0

(

(3.82)


is orthonormal in pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq, ¨L2pBorelp0,1q;|¨|Rq

q. It thus

remains to prove that the set


(

Y


nu, n P N0

(

(3.83)is complete in pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq. To prove this,

we assert that for all m P N it holds that for all l P t1, 2, . . . , 2mu it holds that

1p0,1q,R`

l´12m

, l2m

˘ P spanL2pBorelp0,1q;|¨|Rq

´

1(

Y

Hn,k : k P t1, 2, . . . , 2nu, n P t0, 1, . . . ,m´1u(

¯

.

(3.84)In the following we prove (3.84) by induction on m P N. The base case m “ 1 followsfrom the fact that for all l P t1, 2u it holds that

1p0,1q,R`

l´12, l2

˘ “ 12

”

1p0,1q,Rp0,1q ` p´1qpl´1q

¨H0,1

ı

. (3.85)

The induction step N Q m Ñ m ` 1 P N follows from the induction hypothesis andfrom the fact that for all m P N, l P t1, 2, . . . , 2m`1u it holds that

1p0,1q,R`

l´1

2m`1 ,l

2m`1

˘ “ 12

„

1p0,1q,R`

rl2s´12m

,rl2s

2m

˘ `p´1qpl´1q

2m2¨Hm,rl2s

. (3.86)

Induction hence proves (3.84). Clearly, (3.84) implies (3.83). The proof of Proposi-tion 3.3.22 is thus completed.

3.3.2.3 Transformations of orthonormal bases

Exercise 3.3.23 (Transformation of orthonormal bases). Let a, b, α, β P R witha ă b and α ă β, let en : pa, bq Ñ R, n P N, be functions such that the set ten PL2pBorelpa,bq; |¨|Rq : n P Nu is an orthonormal basis of L2pBorelpa,bq; |¨|Rq, and letfn : pα, βq Ñ R, n P N, be the functions with the property that for all n P N,x P pα, βq it holds that

fnpxq “

d

pb´ aq

pβ ´ αqen

ˆ

px´ αq pb´ aq

pβ ´ αq` a

˙

. (3.87)

Prove that the set tfn P L2pBorelpα,βq; |¨|Rq : n P Nu is an orthonormal basis ofL2pBorelpα,βq; |¨|Rq.


3.4 Linear functions

In the section we particularly follow the presentations in Werner [29].

Definition 3.4.1 (Linear operators). Let K be a field, let V1 and V2 be K-vectorspaces, and let A : V1 Ñ V2 be a function/operator3 with the property that for allv, w P V1, λ P K it holds that

A pλv ` wq “ λAv ` Aw. (3.88)

Then A is called K-linear.

Definition 3.4.2. Let K be a field and let V1 and V2 be K-vector spaces. Then wedenote by LinpV1, V2q the set given by

LinpV1, V2q “ tA PMpV1, V2q : A is linearu (3.89)

(the set of all linear functions from V1 to V2/the set of all linear operators from V1

to V2).

Definition 3.4.3 (Linear operators on a vector space). Let K be a field, let V1 andV2 be K-vector spaces, let U Ď V1 be a vector subspace of V1, and let A P LinpU, V2q.Then A is called a linear operator from U on V1 to V2 (a linear operator on V1 toV2).

Definition 3.4.4 (Set of linear operators on a vector space). Let K be a field andlet V1 and V2 be K-vector spaces. Then we denote by LpV1, V2q the set given by

LpV1, V2q “ď

UĎV1 is a vectorsubspace of V1

LinpU, V2q (3.90)

(the set of linear operators on V1 to V2).

Definition 3.4.5 (Point spectrum of a linear operator). Let K P tR,Cu, let V be aK-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Then we denote byσP pAq the set given by

σP pAq “!

λ P K :`

λ´ A : DpAq Ñ V is not injective˘

)

(3.91)

and we call σP pAq the point spectrum of A (the set of eigenvalues of A).

3A function from some possibly infinite dimensional normed vector space into some possiblyinfinite dimensional normed vector space is also often referred as an “operator”. We will also oftenuse this convention in the remainder of these lecture notes.

3.4. LINEAR FUNCTIONS 63

Definition 3.4.6 (Symmetric linear operators). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator with theproperty that for all v, w P DpAq it holds that

〈Av,w〉H “ 〈v, Aw〉H . (3.92)

Then A is called symmetric.

Definition 3.4.7. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a linear operator.

• A is called nonnegative if

@ v P DpAq : 〈v,Av〉H P r0,8q. (3.93)

• A is called nonpositive if

@ v P DpAq : 〈v, Av〉H P p´8, 0s. (3.94)

• A is called strictly positive if

@ v P DpAqzt0u : 〈v,Av〉H P p0,8q. (3.95)

• A is called strictly negative if

@ v P DpAqzt0u : 〈v, Av〉H P p´8, 0q. (3.96)

3.4.1 Continuous linear functions on normed vector spaces

Definition 3.4.8 (Bounded linear functions/bounded linear operators). Let K P

tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be normed K-vector spaces. Then we denoteby LpV1, V2q the set given by

LpV1, V2q “ LinpV1, V2q X CpV1, V2q (3.97)

(the set of all continuous linear operators from V1 to V2) and we denote by ¨LpV1,V2q :

LpV1, V2q Ñ r0,8q the function with the property that for all A P LpV1, V2q it holdsthat

ALpV1,V2q “ supvPV1zt0u

„

AvV2vV1

. (3.98)


Definition 3.4.9. Let K P tR,Cu and let pV, ¨V q be a normed K-vector space.Then we denote by LpV1q the set given by

LpV1q “ LpV1, V1q (3.99)

(the set of all continuous linear operators from V1 to V1) and we denote by ¨LpV1q : LpV1q Ñ

r0,8q the function with the property that for all A P LpV1q it holds that

ALpV1q “ ALpV1,V1q . (3.100)

Lemma 3.4.10 (Completeness of the space of bounded linear operators). Let K P

tR,Cu, let pV1, ¨V1q be a normed K-vector space, and let pV2, ¨V2q be a K-Banachspace. Then pLpV1, V2q, ¨LpV1,V2qq is a K-Banach space.

Definition 3.4.11 (Topological dual space). Let K P tR,Cu and let pV, ¨V q be anormed K-vector space. Then we denote by pV 1, ¨V 1q the K-Banach space given by

pV 1, ¨V 1q “ pLpV,Kq, ¨LpV,Kqq. (3.101)

See Reed & Simon [25] and, e.g., Prevot & Rockner [24] for the next results.

Theorem 3.4.12 (Square root of a nonnegative and symmetric bounded linearoperator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P

LpHq be nonnegative and symmetric. Then there exists a unique nonnegative andsymmetric S P LpHq with the property that S2 “ A.

Definition 3.4.13 (Square root of a nonnegative and symmetric bounded linearoperator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P

LpHq be nonnegative and symmetric. Then we denote by A12 P LpHq the uniquenonnegative and symmetric bounded linear operator with the property that pA12q2 “

A.

Lemma 3.4.14. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA P LpHq. Then A˚A is nonnegative and symmetric.

Proof of Lemma 3.4.14. Note that for all v P H it holds that

〈v, A˚Av〉H “ 〈Av,Av〉H “ 〈A˚Av, v〉H “ 〈Av,Av〉H “ Av

2H ě 0. (3.102)



3.4.2 Compact operators on Banach spaces

Definition 3.4.15 (Compact operator). LetK P tR,Cu, let pV1, ¨V1q and pV2, ¨V2qbe K-Banach spaces, and let A P LpV1, V2q satisfy that for every bounded subsetB Ď V1 of V1 it holds that ApBq is a relatively compact subset of V2, i.e., it holdsthat ApBq is a compact subset of V2. Then A is called a compact operator.

Definition 3.4.16 (The space of compact operators). Let K P tR,Cu and letpV1, ¨V1q and pV2, ¨V2q be K-Banach spaces. Then we denote by KpV1, V2q the setgiven by

KpV1, V2q “ tA P LpV1, V2q : A is compactu (3.103)

(the set of all compact linear operators from V1 to V2).

Proposition 3.4.17. Let K P tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be K-Banachspaces. Then

tA P LpV1, V2q : dimpimpAqq ă 8uLpV1,V2q

Ď KpV1, V2q. (3.104)

Proposition 3.4.18. Let K P tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be K-Hilbertspaces. Then

tA P LpV1, V2q : dimpimpAqq ă 8uLpV1,V2q

“ KpV1, V2q. (3.105)

3.4.3 Nuclear operators on Banach spaces

3.4.3.1 Definition of Nuclear operators

Definition 3.4.19 (Rank-1 operators). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe normed K-vector spaces, let v P V 1, and let w P W . Then we denote by

`

w bv˘

: V Ñ W the function with the property that for all u P V it holds that

pw b vqpuq “ vpuqw. (3.106)


Definition 3.4.20 (Nuclear operator). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe K-Banach spaces, and let A : V Ñ W be a K-linear operator with the property thatthere exist a sequence pvnqnPN Ď V 1 of elements in V 1 and a sequence pwnqnPN Ď Wof elements in W such that

8ÿ

n“1

vnV 1 wnW ă 8 (3.107)

and such that for all x P V it holds that

Ax “8ÿ

n“1

pwn b vnqpxq “8ÿ

n“1

vnpxqwn. (3.108)

Then A is called a nuclear operator.

Conditions (3.107) and (3.108) say something about how good a linear operatorcan be approximated through sums of linear operators with one-dimensional images.In addition, conditions (3.107) and (3.108) assert that a nuclear operator can bedecomposed into rank one operators in the sense of (3.107)–(3.108). Equation (3.108)is also referred to as a nuclear represenation of a nuclear operator.

Definition 3.4.21 (The normed vector space of nuclear operators). Let K P tR,Cuand let pV, ¨V q and pW, ¨W q be K-Banach spaces. Then we denote by L1pV,W qthe set given by

L1pV,W q “ tA P LinpV,W q : A is nuclearu (3.109)

(the set of all nuclear operators from V to W ) and we denote by ¨L1pV,W q: L1pV,W q Ñ

r0,8q the function with the property that for all A P L1pV,W q it holds that

AL1pV,W q“ inf

#

a P r0,8q :

„

D pvnqnPN Ď V 1 : D pwnqnPN Ď W :

´

a “ř8

n“1 vnV 1 wnW ă 8 and @x P V : Ax “ř8

n“1pwn b vnqpxq¯

+

. (3.110)

Definition 3.4.22. Let K P tR,Cu and let pV, ¨V q be a K-Banach space. Then wedenote by L1pV q the set given by L1pV q “ L1pV, V q and we denote by ¨L1pV q

: L1pV q Ñ

r0,8q the mapping with the property that for all A P L1pV q it holds that AL1pV q“

AL1pV,V q.


3.4.3.2 Relation of bounded linear operators and nuclear operators

Lemma 3.4.23. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaces,and let A P L1pV,W q. Then A P LpV,W q and it holds that

ALpV,W q ď AL1pV,W q. (3.111)

Proof of Lemma 3.4.23. Let ε P p0,8q be arbitrary. The assumption that A P

L1pV,W q ensures that there exist sequences pvnqnPN Ď V 1 and pwnqnPN Ď W suchthat

8ÿ

n“1

vnV 1 wnW ď ε` AL1pV,W qď ε`

8ÿ

n“1

vnV 1 wnW ă 8 (3.112)

and such that for all x P V it holds that

Ax “8ÿ

n“1

vnpxqwn. (3.113)

Then note that for all x P V it holds that

AxW “

›

›

›

›

›

8ÿ

n“1

vnpxqwn

›

›

›

›

›

W

ď

8ÿ

n“1

vnpxqwnW “

8ÿ

n“1

|vnpxq| wnW

ď

8ÿ

n“1

vnLpV,Rq xV wnW “

«

8ÿ

n“1

vnV 1 wnW

ff

xV

ď

”

AL1pV,W q` ε

ı

xV .

(3.114)

This proves that A P LpV,W q and that

ALpV,W q ď AL1pV,W q` ε. (3.115)

As ε P p0,8q was arbitrary, the proof of Lemma 3.4.23 is completed.

Lemma 3.4.23, in particular, proves that in the setting of Lemma 3.4.23 it holdsthat

pL1pV,W q, ¨L1pV,W qq Ď pLpV,W q, ¨LpV,W qq (3.116)

continuously.


3.4.3.3 Structure of the space of nuclear operators

Lemma 3.4.24. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be K-Banach spaces.Then the pair pL1pV,W q, ¨L1pV,W q

q is a normed K-vector space.

Proof of Lemma 3.4.24. Lemma 3.4.23 implies that for all A P L1pV,W q it holdsthat AL1pV,W q

“ 0 if and only if A “ 0. Furthermore, it is clear that for all

A P L1pV,W q, λ P K it holds that λ ¨ A P L1pV,W q and that

λ ¨ AL1pV,W q“ |λ| ¨ AL1pV,W q

. (3.117)

It thus remains to prove that the sum of two nuclear operators from V to W is againa nuclear operator from V to W and that the triangle inequality holds. For this letA1, A2 P L1pV,W q, ε P p0,8q be arbitrary and let vin P V

1, n P N, i P t1, 2u, andwin P W , n P N, i P t1, 2u, satisfy that for all i P t1, 2u it holds that

8ÿ

n“1

›

›vin›

›

V 1

›

›win›

›

Wď ε` AiL1pV,W q

ď ε`8ÿ

n“1

›

›vin›

›

V 1

›

›win›

›

Wă 8 (3.118)

and that for all x P V , i P t1, 2u it holds that

Aix “8ÿ

n“1

vinpxqwin. (3.119)

This implies that8ÿ

n“1

2ÿ

i“1

›

›vin›

›

V 1

›

›win›

›

Wă 8 (3.120)

and that for all x P V it holds that

pA1 ` A2qx “8ÿ

n“1

2ÿ

i“1

vinpxqwin

“ v11pxqw

11 ` v

21pxqw

21 ` v

12pxqw

12 ` v

22pxqw

22 ` v

13pxqw

13 ` v

23pxqw

23 ` . . . .

(3.121)

Hence, we obtain that A1 ` A2 P L1pV,W q and that

A1 ` A2L1pV,W qď

8ÿ

n“1

2ÿ

i“1

›

›vin›

›

V 1

›

›win›

›

Wď A1L1pV,W q

` A2L1pV,W q` 2ε. (3.122)

As ε P p0,8q was arbitrary, the proof of Lemma 3.4.24 is completed.


3.4.3.4 Ideal property of the set of nuclear operators

Proposition 3.4.25. Let K P tR,Cu, let pV0, ¨V0q, pV1, ¨V1q, pW0, ¨W0q and

pW1, ¨W1q be K-Banach spaces, and let A P L1pV1,W1q, B1 P LpW1,W0q, B2 P

LpV0, V1q. Then it holds that B1AB2 P L1pV0,W0q and it holds that

B1AB2L1pV0,W0qď B1LpW1,W0q

AL1pV1,W1qB2LpV1,V0q . (3.123)

Proof of Proposition 3.4.25. Let ε P p0,8q be arbitrary. The assumption that A PL1pV1,W1q ensures that there exist pvnqnPN Ď pV1q

1 and pwnqnPN Ď W1 such that

8ÿ

n“1

vnV 11wnW1

ď ε` AL1pV1,W1qď ε`

8ÿ

n“1

vnV 11wnW1

ă 8 (3.124)

and such that for all x P V1 it holds that

Ax “8ÿ

n“1

vnpxqwn. (3.125)

Inequality (3.124) implies that

8ÿ

n“1

vnpB2p¨qqV 10B1pwnqW0

ď

8ÿ

n“1

vnV 11B2LpV0,V1q B1LpW1,W0q

wnW1

ď B2LpV0,V1q B1LpW1,W0q

”

ε` AL1pV1,W1q

ı

(3.126)

and equation (3.125) ensures that for all x P V0 it holds that

pB1AB2q pxq “8ÿ

n“1

vnpB2pxqqB1pwnq. (3.127)

This implies that B1AB2 is a nuclear operator from V0 to W0 and that

B1AB2L1pV0,W0qď B2LpV0,V1q B1LpW1,W0q

”

ε` AL1pV1,W1q

ı

. (3.128)

As ε P p0,8q was arbitrary, the proof of Proposition 3.4.25 is completed.


3.4.3.5 Characterization of nuclear operators

The next simple lemma gives a characterization for nuclear operators and is animmediate consequence of the definition of a nuclear operator.

Lemma 3.4.26. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaceswith V ‰ t0u and W ‰ t0u, and let A P LpV,W q. Then the following three state-ments are equivalent:

(i) It holds that A P L1pV,W q.

(ii) There exist panqnPN Ď R, pvnqnPN Ď V 1, pwnqnPN Ď W such that for all n P Nit holds that vnV 1 “ wnW “ 1, such that

ř8

n“1 |an| ă 8 and such that forall x P V it holds that

Ax “8ÿ

n“1

an vnpxqwn. (3.129)

(ii) There exist panqnPN Ď r0,8q, pvnqnPN Ď V 1, pwnqnPN Ď W such that for alln P N it holds that vnV 1 “ wnW “ 1, such that

ř8

n“1 an ă 8 and such thatfor all x P V it holds that

Ax “8ÿ

n“1

an vnpxqwn. (3.130)

3.4.4 Hilbert-Schmidt operators on Hilbert spaces

Definition 3.4.27. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q

be K-Hilbert spaces, and let A P LpH1, H2q be a bounded linear operator with theproperty that there exist an orthonormal basis B Ď H1 of H1 such that

ÿ

bPB

Ab2H2ă 8. (3.131)

Then A is called Hilbert-Schmidt.

3.4.4.1 Independence of the orthonormal basis

Lemma 3.4.28. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q beK-Hilbert spaces, let B1 Ď H1 be an orthonormal basis of H1, let B2 Ď H2 be anorthonormal basis of H2, and let A P LpH1, H2q. Then

ÿ

bPB1

Ab2H2“

ÿ

bPB2

A˚b2H1. (3.132)


Proof of Lemma 3.4.28. Observe thatÿ

bPB1

Ab2H2“

ÿ

bPB1

ÿ

bPB2

ˇ

ˇxb, AbyH2

ˇ

ˇ

2“

ÿ

bPB1

ÿ

bPB2

ˇ

ˇxA˚b, byH1

ˇ

ˇ

2

“ÿ

bPB2

ÿ

bPB1

ˇ

ˇxA˚b, byH1

ˇ

ˇ

2“

ÿ

bPB2

A˚b2H1.

(3.133)


Lemma 3.4.29 (Independence of the orthonormal bases). Let K P tR,Cu, letpH1, 〈¨, ¨〉H1

, ¨H1q and pH2, 〈¨, ¨〉H2

, ¨H2q be K-Hilbert spaces, let B1 Ď H1 and

B2 Ď H1 be orthonormal bases of H1, and let A P LpH1, H2q. Thenÿ

bPB1

Ab2H2“

ÿ

bPB2

Ab2H2. (3.134)

Proof of Lemma 3.4.29. Theorem 3.3.6 implies that there exists an orthonormal ba-sis B Ď H2 of H2. Lemma 3.4.28 then implies that

ÿ

bPB1

Ab2H2“

ÿ

bPB

A˚b2H1“

ÿ

bPB2

Ab2H2. (3.135)


The next result, Corollary 3.4.30, gives a characterization of Hilbert-Schmidtoperators and follows immediately from Lemma 3.4.29 above.

Corollary 3.4.30. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q

be K-Hilbert spaces, and let A P LpH1, H2q. Then A is a Hilbert-Schmidt operator ifand only if for every orthonormal basis B Ď H1 of H1 it holds that

ÿ

bPB

Ab2H1ă 8. (3.136)

3.4.4.2 The Hilbert space of Hilbert-Schmidt operators

Definition 3.4.31 (The space of Hilbert-Schmidt operators). Let K P tR,Cu andlet pH1, 〈¨, ¨〉H1

, ¨H1q and pH2, 〈¨, ¨〉H2

, ¨H2q be K-Hilbert spaces. Then we denote

by L2pH1, H2q and HSpH1, H2q the set given by

L2pH1, H2q “ HSpH1, H2q “ tA P LpH1, H2q : A is Hilbert-Schmidtu (3.137)

(the set of all Hilbert-Schmidt operators from H1 to H2).


In the next definition, Definition 3.4.32, we introduce a norm on a space of Hilbert-Schmidt operators. Lemma 3.4.29 above ensures that (3.138) in Definition 3.4.32 doesindeed make sense.

Definition 3.4.32 (A norm on the space of Hilbert-Schmidt operators). Let K P

tR,Cu and let pH1, 〈¨, ¨〉H1, ¨H1

q and pH2, 〈¨, ¨〉H2, ¨H2

q be K-Hilbert spaces. Thenwe denote by ¨L2pH1,H2q

“ ¨HSpH1,H2q: L2pH1, H2q Ñ r0,8q the function with the

property that for all A P L2pH1, H2q and all orthonormal bases B Ď H1 of H1 it holdsthat

AL2pH1,H2q“ AHSpH1,H2q

“

«

ÿ

bPB

Ab2H2

ff12

. (3.138)

3.4.4.3 Hilbert-Schmidt embeddings

Lemma 3.4.33 (Hilbert-Schmidt embeddings). Let rn P Rzt0u, n P N, be realnumbers with

ř8

n“11

|rn|2ă 8, let pH, 〈¨, ¨〉H , ¨Hq and pH, 〈¨, ¨〉H , ¨Hq be R-Hilbert

spaces with H Ď H continuously, and let en P H, n P N, be an orthonormal basis ofH which satisfies that rnen, n P N, is an orthonormal basis of H. Then

〈v, w〉H “8ÿ

n“1

〈en, v〉H 〈en, w〉H|rn|

2 . (3.139)

Proof. Note that for all n P N it holds that

〈en, v〉H “

⟨en,

8ÿ

m“1

〈em, v〉H em

⟩H

“

8ÿ

m“1

〈em, v〉H 〈en, em〉H

“ 〈en, v〉H 〈en, en〉H “ 〈en, v〉H en2H “

〈en, v〉H|rn|2

¨ rnen2H “

〈en, v〉H|rn|2

.

(3.140)

This implies that

〈v, w〉H “8ÿ

n“1

〈rnen, v〉H 〈rnen, w〉H “8ÿ

n“1

〈en, v〉H 〈en, w〉H|rn|

2 . (3.141)


3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 73

3.5 Diagonal linear operators on Hilbert spaces

Definition 3.5.1 (Diagonal linear operators). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator with theproperty that there exists an orthonormal basis B Ď H of H and a function λ : BÑ K

such that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(3.142)

and such that for all v P DpAq it holds that

Av “ÿ

bPB

λb 〈b, v〉H b. (3.143)

Then we say that A is a diagonal linear operator.

Exercise 3.5.2 (Diagonal operators are densely defined). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonal

linear operator. Prove that A is densely defined, i.e., prove that DpAq “ H.

Exercise 3.5.3. Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let A : DpAq Ď H Ñ

H be a diagonal linear operator. Prove that A is symmetric.

Exercise 3.5.4 (The point spectrum of a diagonal operator). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator, i.e., assume that there exists an orthonormal basis B Ď H of H anda function λ : BÑ K such that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(3.144)

and such that for all v P DpAq it holds that Av “ř

bPB λb 〈b, v〉H b. Prove thatσP pAq “ impλq.


Proposition 3.5.5 (Regularity of diagonal linear operators). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linear operator,let B Ď H be an orthonormal basis of H, and let λ : BÑ K be a function such that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(3.145)

and such that for all v P DpAq it holds that Av “ř

bPB λb 〈b, v〉H b. Then

(i) A P LpHq if and only if λ P L8p#B; |¨|Kq (ô λ is a bounded function ô

impλq “ σP pAq is a bounded set) and in that case it holds that ALpHq “

λL8p#B;|¨|Kq“ supbPB |λb|,

(ii) A P L2pHq “ HSpHq if and only if λ P L2p#B; |¨|Kq (ô

ř

bPB |λb|2ă 8) and

in that case it holds that AL2pHq“ λL2p#B;|¨|

Kq““ř

bPB |λb|2‰12

, and

(iii) A P L1pHq if and only if λ P L1p#B; |¨|Kq (ô

ř

bPB |λb| ă 8) and in that caseit holds that AL1pHq

“ λL1p#B;|¨|Kq“ř

bPB |λb|.

Question 3.5.6. Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let b : N Ñ H be aninjective mapping such that bpNq “ tb1, b2, . . . u Ď H is an orthonormal basis of H,let Ar : DpArq Ď H Ñ H, r P R, be linear operators with the property that for allr P R it holds that DpArq “

v P H :ř8

n“1 n2r |〈bn, v〉H |

2ă 8

(

and with the propertythat for all r P R, v P DpArq it holds that

Arv “8ÿ

n“1

nr 〈bn, v〉H bn. (3.146)

Specify three subsets S1 Ď R, S2 Ď R and S3 Ď R of the real numbers

• such that for all r P R it holds that Ar P HSpHq if and only if r P S1,

• such that for all r P R it holds that Ar P LpHq if and only if r P S2, and

• such that for all r P R it holds that Ar P L1pHq if and only if r P S3.

3.5.1 Laplace operators on bounded domains

In this section we give functional analytic descriptions of Laplace operators with suit-able boundary conditions and thereby present a few important examples of diagonallinear operators.


3.5.1.1 Laplace operators with Dirichlet boundary conditions

Definition 3.5.7 (Laplace operator with Dirichlet boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all x P

p0, 1q it holds that enpxq “?

2 sinpnπxq, and let A : DpAq Ď L2pBorelp0,1q; |¨|Rq ÑL2pBorelp0,1q; |¨|Rq be the linear operator with the property that

DpAq “

#

v P L2pBorelp0,1q; |¨|Rq :

8ÿ

n“1

n4ˇ

ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq

ˇ

ˇ

2

Ră 8

+

(3.147)

and with the property that for all v P DpAq it holds that

Av “8ÿ

n“1

´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.148)

Then we refer to A as the Laplace operator with Dirichlet boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.5.8. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t´π

2, ´4π2, ´9π2,´16π2, . . . u, it holds that

DpAq “ H2pp0, 1q;Rq

loooooomoooooon

Sobolev space

XH10 pp0, 1q;Rq

loooooomoooooon

Sobolev space

“

$

’

’

’

&

’

’

’

%

v P H2pp0, 1q;Rq : lim

xŒ0vpxq

looomooon

“vp0`q

“ limxÕ1

vpxqlooomooon

“vp1´q

“ 0

,

/

/

/

.

/

/

/

-

,

(3.149)

and it holds for all v P DpAq that Av “ v2.


3.5.1.2 Laplace operators with Neumann boundary conditions

Definition 3.5.9 (Laplace operator with Neumann boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P N0, satisfy that for all n P N and Borelp0,1q-almost all

x P p0, 1q it holds that e0pxq “ 1 and enpxq “?

2 cospnπxq, and let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with the property that

DpAq “

#

v P L2pBorelp0,1q; |¨|Rq :

8ÿ

n“1

n4ˇ


ˇ

ˇ

2

Ră 8

+

(3.150)


Av “8ÿ

n“0

´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.151)

Then we refer to A as the Laplace operator with Neumann boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.5.10. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Neumann boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´π2, ´4π2,´9π2, ´16π2, . . . u, it holds that

DpAq “

$

’

’

’

&

’

’

’

%


xŒ0v1pxq

looomooon

“v1p0`q

“ limxÕ1

v1pxqlooomooon

“v1p1´q

“ 0

,

/

/

/

.

/

/

/

-

, (3.152)



3.5.1.3 Laplace operators with periodic boundary conditions

Definition 3.5.11 (Laplace operator with periodic boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P Z, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q

it holds that e0pxq “ 1, enpxq “?

2 sinp2nπxq and e´npxq “?

2 cosp2nπxq, and letA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with theproperty that

DpAq “

#

v P H :ÿ

nPZ

n4ˇ


ˇ

ˇ

2

Ră 8

+

(3.153)


Av “ÿ

nPZ

´4π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.154)

Then we refer to A as the Laplace operator with periodic boundary conditions onL2pBorelp0,1q; |¨|Rq.

Proposition 3.5.12. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with periodic boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´4¨π2, ´4¨22 ¨π2,´4 ¨ 32 ¨ π2, ´4 ¨ 42 ¨ π2, . . . u, it holds that

DpAq “ H2P pp0, 1q;Rq

loooooomoooooon

Sobolev space

“

$

’

’

’

&

’

’

’

%


xŒ0vpxq

looomooon

“vp0`q

“ limxÕ1

vpxqlooomooon

“vp1´q

, limxŒ0

v1pxqlooomooon

“v1p0`q

“ limxÕ1

v1pxqlooomooon

“v1p1´q

,

/

/

/

.

/

/

/

-

,

(3.155)



3.5.2 Spectral decomposition for a diagonal linear operator

Proposition 3.5.13 (The eigenspaces of diagonal linear operators). Let K P tR,Cube a field, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator, i.e., assume that there exists an orthonormal basis B Ď Hand a function λ : BÑ K such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(3.156)


Av “ÿ

bPB

λb 〈b, v〉H b. (3.157)

Then for all µ P σP pAq it holds that

Kernpµ´ Aq “ spantb P B : λb “ µu “ spanpλ´1pµqq (3.158)

Proof of Proposition 3.5.13. Let µ P σP pAq be arbitrary. We first prove that

Kernpµ´ Aq Ď spantb P B : λb “ µu. (3.159)

For this observe that for all

v P Kernpµ´ Aq “ tw P Dpµ´ Aq “ DpAq : pµ´ Aqw “ 0u (3.160)

it holds that

0 “ 02H “ pµ´ Aqv2H “

ÿ

bPB

|pµ´ λbq 〈b, v〉H |2“

ÿ

bPB

|µ´ λb|2|〈b, v〉H |

2

“ÿ

bPB,λb‰µ

|µ´ λb|2|〈b, v〉H |

2 .(3.161)

Hence, we obtain that for all v P Kernpµ ´ Aq and all b P B with λb ‰ µ it holdsthat 〈b, v〉H “ 0. This implies that for all v P Kernpµ´ Aq it holds that

v P”

spantb P B : λb ‰ µuıK

. (3.162)

This and the identity”

spantb P B : λb ‰ µuı

k

”

spantb P B : λb “ µuı

“ H (3.163)


prove that (3.159) is indeed fufilled. Next we prove that

Kernpµ´ Aq Ě spantb P B : λb “ µu. (3.164)

For this observe that for all

v P spantb P B : λb “ µu “”


(3.165)

it holds thatÿ

bPB

|λb 〈b, v〉H |2“

ÿ

bPBλb“µ

|λb 〈b, v〉H |2“ |µ|2

ÿ

bPBλb“µ

|〈b, v〉H |2“ |µ|2 v2H ă 8. (3.166)

This shows that

spantb P B : λb “ µu “”


Ď DpAq “ Dpµ´ Aq. (3.167)

Next note that for all

v P spantb P B : λb “ µu “”


(3.168)

it holds that

pµ´ Aq v “ÿ

bPB

pµ´ λbq 〈b, v〉H b “ÿ

bPBλb“µ

pµ´ λbq 〈b, v〉H b “ 0. (3.169)


The next result, Theorem 3.5.14, establishes a spectral decomposition for diagonallinear operators. It follows immediately from Proposition 3.5.13 above.

Theorem 3.5.14 (Spectral decomposition for diagonal linear operators). Let K P

tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator. Then it holds that

DpAq “

$

&

%

v P H :ÿ

λPσP pAq

|λ|2›

›PKernpλ´Aqpvq›

›

2

Hă 8

,

.

-

(3.170)

and it holds for all v P DpAq that

Av “ÿ

λPσP pAq

λ ¨ PKernpλ´Aqpvq. (3.171)

Exercise 3.5.15. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a diagonal linear operator. Prove that A is symmetric if andonly if σP pAq Ď R.


3.5.3 Fractional powers of a diagonal linear operator

Definition 3.5.16 (Nonnegative fractional powers of a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P r0,8q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď r0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλÁqpvq›

›

2

Hă 8

,

.

-

(3.172)

and with the property that for all v P DpArq it holds that

Arv “ÿ

λPσP pAq

λr ¨ PKernpλÁqpvq. (3.173)

Definition 3.5.17 (Negative fractional powers of a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P p´8, 0q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that

DpArq “

$

&

%

v P H :ÿ

λPσP pAq

›

›λr ¨ PKernpλÁqpvq›

›

2

Hă 8

,

.

-

(3.174)

and with the property that for all v P DpArq it holds that

Arv “ÿ

λPσP pAq

λr ¨ PKernpλÁqpvq. (3.175)

The next lemma collects a simple property of fractional powers of a diagonallinear operator. It follows immediately from Definition 3.5.16 and Definition 3.5.17.

Lemma 3.5.18 (Diagonality of fractional powers of a diagonal linear operators).Let K P tR,Cu, r P R, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq ĎH Ñ H be a diagonal linear operator with σP pAq Ď r0,8q, and assume that

`

r Pr0,8q or σP pAq Ď p0,8q

˘

. Then Ar is a diagonal linear operator.


3.5.4 Domain Hilbert space associated to a diagonal linearoperator

Lemma 3.5.19. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Thenthe triple

`

DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘

is a K-inner product space.

The proof of Lemma 3.5.19 is clear and therefore omitted. If the point spectrum ofthe diagonal linear operator A in Lemma 3.5.19 in addition satisfies infpσP pAqq ą 0,then the triple

`


is even a K-Hilbert space. This is thesubject of the next lemma.

Lemma 3.5.20. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q andinfpσP pAqq ą 0. Then the triple

`


is a K-Hilbert spaceand it holds for all v P DpAq that

vH ďAvH

infpσP pAqq. (3.176)

Proof of Lemma 3.5.20. First of all, note that for all v P DpAq it holds that

Av2H “ÿ

µPσP pAq

›

›µ ¨ PKernpµÁqpvq›

›

2

H“

ÿ

µPσP pAq

|µ|2 ¨›

›PKernpµÁqpvq›

›

2

H

ě

„

infµPσP pAq

|µ|2

»

–

ÿ

µPσP pAq

›

›PKernpµÁqpvq›

›

2

H

fi

fl “

„

infµPσP pAq

µ

2

v2H .

(3.177)

This proves (3.176). Due to Lemma 3.5.19, it remains to prove that the normedK-vector space

`

DpAq, Ap¨qH˘

is complete. For this let pvnqnPN Ď DpAq be aCauchy sequence in

`

DpAq, Ap¨qH˘

. Inequality (3.176) hence implies that pvnqnPNis a Cauchy sequence in pH, ¨Hq too. This and the fact that pH, ¨Hq is completeshows that there exists a vector v P H such that limnÑ8 vn ´ vH “ 0. Next note


that for all n P N it holds that

lim infmÑ8

Apvn ´ vmq2H “ lim inf

mÑ8

»

–

ÿ

µPσP pAq

|µ|2›

›PKernpµÁqpvn ´ vmq›

›

2

H

fi

fl

ěÿ

µPσP pAq

|µ|2”

lim infmÑ8

›

›PKernpµÁqpvn ´ vmq›

›

2

H

ı

“ÿ

µPσP pAq

|µ|2„

›

›

›PKernpµÁq

´

vn ´ limmÑ8

vm

¯›

›

›

2

H

“ÿ

µPσP pAq

|µ|2”

›

›PKernpµÁqpvn ´ vq›

›

2

H

ı

.

(3.178)

This proves that v P DpAq and that limnÑ8 Apvn ´ vqH “ 0. The proof ofLemma 3.5.20 is thus completed.

Exercise 3.5.21. Give an example of an R-Hilbert space pH, 〈¨, ¨〉H , ¨Hq and adiagonal linear operator A : DpAq Ď H Ñ H such that σP pAq Ď p0,8q and such thatthe triple

`


is not an R-Hilbert space.

3.5.5 Interpolation spaces associated to a diagonal linear op-erator

Theorem 3.5.22 (Completion). Let pE, dEq be a metric space. Then there exists a

complete metric space pF, dF q such that E Ď F , EF“ F and dF |EÊ “ dE.

Theorem 3.5.22 can be proved by considering the set of equivalence classes ofCauchy sequences in E. The detailed proof of Theorem 3.5.22 is well known andtherefore omitted.

Definition 3.5.23 (Completion). Let pE, dEq be a metric space and let pF, dF q be a

complete metric space such that E Ď F , EF“ F and dF |EÊ “ dE. Then the metric

space pF, dF q is called a completion of pE, dEq.

We now introduce the concept of a family of interpolation spaces associated to adiagonal linear operator.


Theorem 3.5.24 (Interpolation spaces associated to a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0. Then there exists anup to isometric isomorphisms unique family pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, of K-Hilbertspaces with the property that

(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs

,

(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and

(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .

Proof of Theorem 3.5.24. Let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P r0,8q, be theK-Hilbert spaceswith the property that for all r P r0,8q it holds that

pHr, 〈¨, ¨〉Hr , ¨Hrq “ pDpArq, 〈Arp¨q, Arp¨q〉H , A

rp¨qHq. (3.179)

Note that Lemma 3.5.20 ensures that such K-Hilbert spaces do indeed exist. In thenext step let H8 be the set given by H8 “ XrPp0,8qHr. Then we observe that H8is a K-vector space. Furthermore, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P p´8, 0q, be K-Hilbertspaces with the property that for all r P p´8, 0q it holds that

Hr “

H Z

#

pH,ϕq P tHu ˆ LinpH8,Kq :

«

supvPH8zt0u

|ϕpvq|

vH´ră 8 “ sup

vPH8zt0u

|ϕpvq|

vH

ff+

,

(3.180)

with the property that for all r P p´8, 0q, λ P K, v, w P H it holds that

vHr “ supuPH8zt0u

|〈v, u〉H |uH´r

, v `Hr w “ v ` w, λ ¨Hr v “ λ ¨ v, (3.181)

with the property that for all r P p´8, 0q, λ P K, ϕ P LinpH8,Kq with supvPH8zt0u|ϕpvq|vH´r

ă 8 it holds that

pH,ϕqHr “ supuPH8zt0u

|ϕpuq|

uH´r, λ ¨Hr pH,ϕq “ pH, λ ¨ ϕq, (3.182)


and with the property that for all r P p´8, 0q, v P H, ϕ, ψ P LinpH8,Kq with

supvPH8zt0u|ϕpvq|`|ψpvq|vH´r

ă 8 it holds that

pH,ϕq `Hr pH,ψq

“

#

w : rDw P H : @u P H8 : 〈w, u〉H “ ϕpuq ` ψpuqs

pH,ϕ` ψq : else

(3.183)

and

pH,ϕq `Hr v “ v `Hr pH,ϕq “´

H,“

H8 Q u ÞÑ 〈v, u〉H ` ϕpuq P K‰

¯

. (3.184)


Definition 3.5.25 (Interpolation spaces associated to a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H bea symmetric diagonal linear operator with infpσP pAqq ą 0, and let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be K-Hilbert spaces with the property that

(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs

,

(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and

(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .

Then we call pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, a family of interpolation spaces associatedto A.

3.6 The Bochner integral

3.6.1 Existence and uniqueness of the Bochner integral

Theorem 3.6.1 (Bochner integral). Let pΩ,F , µq be a finite measure space, let K PtR,Cu, and let pV, ¨V q be a K-Banach space. Then

(i) there exists a unique continuous K-linear function I : L1pµ; ¨V q Ñ V with theproperty that for all F/BpV q-simple f : Ω Ñ V it holds that

Ipfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v (3.185)

(ii) and it holds for all f P L1pµ; ¨V q that IpfqV ď fL1pµ;¨V q.

3.6. THE BOCHNER INTEGRAL 85

Proof of Theorem 3.6.1. Throughout this proof let S Ď L1pµ; ¨V q be the set of allF/BpV q-simple functions and let J : S Ñ V be the mapping with the property thatfor all f P S it holds that

Jpfq “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.186)

Next observe that the triangle inequality proves that for all f P S it holds that

JpfqV ďř

vPfpΩq

µpf´1pvqq ¨ vV “ fL1pµ;¨V q. (3.187)

This, the fact that J is linear, and Lemma 2.5.22 imply that J is uniformly contin-uous. In addition, we note that item (iv) in Theorem 2.4.7 and Lebesgue’s theoremof dominated convergence ensure that SL1pµ;¨V q “ L1pµ; ¨V q. The assumption thatV is complete hence allows us to apply Proposition 2.5.19 to obtain that there existsa unique I P CpL1pµ; ¨V q, V q with the property that I|S “ J . This proves item (i).

In addition, observe that item (i), (3.187), and the fact that SL1pµ;¨V q “ L1pµ; ¨V qestablish (ii). The proof of Theorem 3.6.1 is thus completed.

3.6.2 Definition of the Bochner integral

Definition 3.6.2. Let pΩ,F , µq be a finite measure space, let K P tR,Cu, and letpV, ¨V q be a K-Banach space. Then we denote by

ş

Ωp¨q dµ : L1pµ; ¨V q Ñ V the

continuous K-linear function with the property that for all F/BpV q-simple f : Ω Ñ Vit holds that

ż

Ω

f dµ “ř

vPfpΩq

µpf´1ptvuqq ¨ v. (3.188)

Corollary 3.6.3 (Triangle inequality for the Bochner integral). Let pΩ,F , µq be afinite measure space, let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letf P L1pµ; ¨V q. Then

›

›

›

›

ż

Ω

f dµ

›

›

›

›

V

ď

ż

Ω

fV dµ. (3.189)

Corollary 3.6.3 is an immediate consequence of Theorem 3.6.1.

Chapter 4

Semigroups of bounded linearoperators

In this chapter we follow with some minor changes the presentations in Pazy [23].

4.1 Definition of a semigroup of bounded linear

operators

Definition 4.1.1 (Semigroups of bounded linear operators). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let S : r0,8q Ñ LpV q be a mapping withthe property that for all t1, t2 P r0,8q it holds that

S0 “ IdV and St1St2 “ St1`t2looooooomooooooon

semigroup property

. (4.1)

Then we call S a semigroup (of bounded linear operators on V ).

4.2 Types of semigroups

Definition 4.2.1 (Contraction semigroups). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with the prop-erty that

suptPr0,8q

StLpV q ď 1. (4.2)

Then S is called a contraction semigroup.

87

88 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS

Definition 4.2.2 (Strongly continuous semigroups). Let K P tR,Cu, let pV, ¨V q bea normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with the propertythat for all v P V it holds that the function

r0,8q Q t ÞÑ Stv P V (4.3)

is continuous. Then S is called strongly continuous.

Definition 4.2.3 (Uniformly continuous semigroups). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with theproperty that the function

r0,8q Q t ÞÑ St P LpV q (4.4)

is continuous. Then S is called uniformly continuous.

Example 4.2.4 (Matrix exponential). Let d P N and let A P Rdˆd be an arbitraryd ˆ d-matrix. Then the function r0,8q Q t ÞÑ eAt P Rdˆd is a uniformly continuoussemigroup.

Clearly, it holds that every uniformly continuous semigroup is also strongly con-tinuous. However, not every strongly continuous semigroup is uniformly continuoustoo. This is the subject of the next exercise.

Exercise 4.2.5. Give an example of a normed R-vector space pV, ¨V q and a stronglycontinuous semigroup S : r0,8q Ñ LpV q so that S is not a uniformly continuoussemigroup. Prove that your function S does indeed fulfill the desired properties.

4.3 The generator of a semigroup

Definition 4.3.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, andlet S : r0,8q Ñ LpV q be a semigroup. Then we denote by GS : DpGSq Ď V Ñ V thefunction with the property that

DpGSq “"

v P V : limtŒ0

„

Stv ´ v

t

exists

*

(4.5)

and with the property that for all v P DpGSq it holds that

GSv “ limtŒ0

„

Stv ´ v

t

(4.6)

and we call GS the (infinitesmal) generator of S.

4.3. THE GENERATOR OF A SEMIGROUP 89

Lemma 4.3.2 (Invariance of the domain of the generator). Let K P tR,Cu, letpV, ¨V q be a normed vector space, and let S : r0,8q Ñ LpV q be a semigroup. Thenit holds for all t P r0,8q that

St`

DpGSq˘

Ď DpGSq (4.7)

and it holds for all t P r0,8q, v P DpGSq that

GSStv “ StGSv. (4.8)

Proof of Lemma 4.3.2. Observe that for all t P r0,8q, v P DpGSq it holds that

limsŒ0

„

Ss rStvs ´ rStvs

s

“ limsŒ0

„

St

„

Ssv ´ v

s

“ St

„

limsŒ0

„

Ssv ´ v

s

“ StGSv. (4.9)

This completes the proof of Lemma 4.3.2.

In the next notion we label all linear operators that are generators of stronglycontinuous semigroups.

Definition 4.3.3 (Generator of a strongly continuous semigroup). Let K P tR,Cu,let pV, ¨V q be a K-Banach space, and let A : DpAq Ď V Ñ V be a linear operatorwith the property that there exists a strongly continuous semigroup S : r0,8q Ñ LpV qsuch that

GS “ A. (4.10)

Then we say that A is a generator of a strongly continuous semigroup.

We complete this section with a simple exercise which aims to illustrate and relatethe different concepts that have been introduced so far in this chapter.

Exercise 4.3.4. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space withV ‰ t0u, and let S : r0,8q Ñ LpV q be the function with the property that for allt P r0,8q it holds that

St “

#

IdV : t “ 0

0 : t ą 0. (4.11)

(i) Is S a semigroup? Prove that your answer is correct.

(ii) Is S a strongly continuous semigroup? Prove that your answer is correct.

(iii) Is S a uniformly continuous semigroup? Prove that your answer is correct.

(iv) Is S a contraction semigroup? Prove that your answer is correct.

(v) Specify DpGSq and GS.


4.4 A global a priori bound for semigroups

In the next result, Proposition 4.4.1, we present a global a priori bound for semigroupsof bounded linear operators.

Proposition 4.4.1 (A global a priori bound). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then it holds forall t P r0,8q, ε P p0,8q that

supsPr0,ts StLpV q ď“

supsPr0,εs SsLpV q‰

¨ et“

lnpSε1εLpV qq

‰`

. (4.12)

Proof of Proposition 4.4.1. Note that for all t P r0,8q, ε P p0,8q, n P N0Xptε´1, tεsit holds that

StLpV q “›

›Snε`ptńεq›

›

LpV q“›

›SnεSptńεq›

›

LpV qď SnεLpV q

›

›Sptńεq›

›

LpV q

“ rSεsnLpV q

›

›Sptńεq›

›

LpV qď Sε

nLpV q

›

›Sptńεq›

›

LpV q

ď“


SεnLpV q ď

“

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰n

ď“

supsPr0,εs SsLpV q‰ “

max

1, SεLpV q(‰tε

““


max!

e0, exp´

t ln´

Sε1εLpV q

¯¯)

ď“


exp´

tmax

0, ln`

Sε1εLpV q

˘(

¯

.

(4.13)

This completes the proof of Proposition 4.4.1.

4.5 Strongly continuous semigroups

4.5.1 A priori bounds for strongly continuous semigroups

In Corollary 4.5.3 below we present a global priori bound for strongly continuoussemigroups. The proof of Corollary 4.5.3 uses the local a priori bound in Lemma 4.5.2below. The proof of Lemma 4.5.2 makes use of the uniform boundedness principle.This is the subject of the next result.

4.5. STRONGLY CONTINUOUS SEMIGROUPS 91

Theorem 4.5.1 (Uniform boundedness principle). Let K P tR,Cu, let pU, ¨Uq bea K-Banach space, let pV, ¨V q be a normed K-vector space, and let A Ď LpU, V q bea set with the property that for all u P U it holds that

supAPA

AuV ă 8. (4.14)

ThensupAPA

ALpU,V q ă 8. (4.15)

Lemma 4.5.2 (Local a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banachspace, and let S : r0,8q Ñ LpV q be a semigroup which satisfies that for all v P V itholds that limtŒ0 Stv “ v. Then

lim suptŒ0

StLpV q “ limtŒ0

supsPr0,ts

SsLpV q ă 8. (4.16)

Proof of Lemma 4.5.2. We prove Lemma 4.5.2 by a contradiction. More specifically,we assume in the following that

limtŒ0

supsPr0,ts

SsLpV q “ 8. (4.17)

This and the fact that S0LpV q “ 1 ă 8 imply that for all t P p0,8q it holds that

supsPp0,ts

SsLpV q “ 8. (4.18)

Hence, there exists a strictly decreasing sequence tn P p0,8q, n P N, with limnÑ8 tn “0 and with the property that for all n P N it holds that

StnLpV q ě n. (4.19)

This ensures thatsupnPN

StnLpV q “ 8. (4.20)

Theorem 4.5.1 hence implies that there exists a vector v P V such that

supnPN

StnvV “ 8. (4.21)

Combining this and the fact that @n P N : StnvV ă 8 implies that

lim supnÑ8

StnvV “ 8. (4.22)


This and the assumption that @ v P V : limtŒ0 Stv “ v show that

8 ą vV “›

›

›limnÑ8

rStnvs›

›

›

V“ lim

nÑ8StnvV “ lim sup

nÑ8StnvV “ 8. (4.23)

This contradiction completes the proof of Lemma 4.5.2.

The next result, Corollary 4.5.3, proves a stronger version of Lemma 4.5.2. Ob-serve that Lemma 4.5.2 and Corollary 4.5.3 apply to strongly continuous semigroupson Banach spaces.

Corollary 4.5.3 (Global a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup which satisfies that for allv P V it holds that limtŒ0 Stv “ v. Then it holds for all t P r0,8q, ε P p0,8q that

supsPr0,ts SsLpV q ď“


¨ et“

lnpSε1εLpV qq

‰`

ă 8. (4.24)

Corollary 4.5.3 is an immediate consequence of Proposition 4.4.1 and Lemma 4.5.2above.

4.5.2 Pointwise convergence in the space of bounded linearoperators

Lemma 4.5.4 (A characterization of pointwise convergence in the space of boundedlinear operators). LetK P tR,Cu, let pV, ¨V q be aK-Banach space, and let pSnqnPN0 Ď

LpV q. Then @ v P V : limnÑ8 Snv ´ S0vV “ 0 if and only if for all compact setsK Ď V it holds that limnÑ8 supvPK Snv ´ S0vV “ 0.

Proof of Lemma 4.5.4. The proof of the “ð” direction in the statement of Lemma 4.5.4is clear. It thus remains to prove the “ñ” direction in the statement of Lemma 4.5.4.To this end we assume that for all v P V it holds that limnÑ8 Snv “ S0v and weassume that there exists a compact set K Ď V such that

lim supnÑ8

supvPK

Snv ´ S0vV ą 0. (4.25)

In the next step we note that there exists a sequence pvnqnPN Ď K such that for alln P N it holds that

Snvn ´ S0vnV “ supvPK

Snv ´ S0vV . (4.26)


The compactness of K ensures that there exist a w P K and a strictly increasingsequence pnkqkPN Ď N such that limkÑ8 vnk “ w. By assumption it holds thatlimkÑ8 Snkw “ S0w. This and Theorem 4.5.1 imply that

0 “ lim supkÑ8

Snkw ´ S0wV

“ lim supkÑ8

Snkpw ´ vnkq ` pSnk ´ S0q vnk ` S0pvnk ´ wqV

ě lim supkÑ8

pSnk ´ S0q vnkV ´ lim supkÑ8

Snkpw ´ vnkqV ´ limkÑ8

S0 pvnk ´ wqV

ě lim supkÑ8

supvPK

pSnk ´ S0q vV ´

„

supkPN

SnkLpV q

lim supkÑ8

w ´ vnkV

“ lim supkÑ8

supvPK

pSnk ´ S0q vV ą 0.

(4.27)

This condradiction completes the proof of Lemma 4.5.4.

4.5.3 Existence of solutions of linear ordinary differentialequations in Banach spaces

Lemma 4.5.5. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, let S : r0,8q ÑLpV q be a strongly continuous semigroup, and let v P DpGSq. Then the functionr0,8q Q t ÞÑ Stv P V is continuously differentiable and it holds for all t P r0,8q that

ddtrStvs “ GSStv “ StGSv. (4.28)

Proof of Lemma 4.5.5. Observe that for all s, t P r0,8q with s ‰ t it holds that

›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

“

›

›

›

›

Sminps,tq

„

Ss´minps,tqv ´ St´minps,tqv

s´ t

´ GSStv›

›

›

›

V

ď

›

›

›

›

Sminps,tq

„

Smaxps,tq´minps,tqv ´ v

maxps, tq ´minps, tq´ GSv

›

›

›

›

V

`›

›

“

Sminps,tq ´ St‰

GSv›

›

V

ď›

›Sminps,tq

›

›

LpV q

›

›

›

›



›

›

›

›

V

`›

›

“

Sminps,tq ´ St‰

GSv›

›

V.

(4.29)


Corollary 4.5.3 and the fact that S is strongly continuous hence imply that for allt P r0,8q it holds that

limr0,8qQsÑt

›

›

›

›

Ssv ´ Stv

s´ t´ StGSv

›

›

›

›

V

ď

«

supsPr0,t`1s

SsLpV q

ff

„

limr0,8qQsÑt

›

›

›

›



›

›

›

›

V

` limr0,8qQsÑt

›

›

“

Sminps,tq ´ St‰

GSv›

›

V“ 0.

(4.30)

This and Lemma 4.3.2 complete the proof of Lemma 4.5.5.

4.5.4 Domains of generators of strongly continuous semi-groups

In this subsection we prove that the generator of a strongly continuous semigroupis densily defined ; see Corollary 4.5.7 below. In the proof of Corollary 4.5.7 we usethe following result, Lemma 4.5.6. Lemma 4.5.6 and its proof can, e.g., be found asTheorem 2.4 (b) in Pazy [23] and Corollary 4.5.7 and its proof can, e.g., be found asCorollary 2.5 in Pazy [23].

Lemma 4.5.6 (Fundamental theorem of calculus for strongly continuous semi-groups). LetK P tR,Cu, t P r0,8q, let pV, ¨V q be aK-Banach space, let S : r0,8q Ñ

LpV q be a strongly continuous semigroup, and let v P V . Then it holds thatşt

0Ssv ds P

DpGSq and it holds that

GSˆż t

0

Ssv ds

˙

“ Stv ´ v. (4.31)

Proof of Lemma 4.5.6. Throughout this proof we assume w.l.o.g. that t P p0,8q.Then we observe that for all u P p0, tq it holds that

rSu ´ IdV s

u

„ż t

0

Ssv ds

“1

u

ż t

0

rSu`sv ´ Ssvs ds “1

u

ż t`u

t

Ssv ds´1

u

ż u

0

Ssv ds.

(4.32)Continuity of the function r0,8q Q s ÞÑ Ssv P V hence proves that

şt

0Ssv ds P DpGSq

and that

GSˆż t

0

Ssv ds

˙

“ limuŒ0

„

rSu ´ IdV s

u

„ż t

0

Ssv ds

“ Stv ´ S0v “ Stv ´ v. (4.33)



We are now ready to prove that the generator of a strongly continuous semigroupis densily defined.

Corollary 4.5.7. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letS : r0,8q Ñ LpV q be a strongly continuous semigroup. Then DpGSq is dense inV .

Proof of Corollary 4.5.7. Let v P V be arbitrary. The assumption that S is a stronglycontinuous semigroup together with the fundamental theorem of calculus ensures that

limtŒ0

ˆ

1

t

ż t

0

Ssv ds

˙

“ v. (4.34)

In addition, Lemma 4.5.6 proves that for all t P p0,8q it holds that 1t

şt

0Ssv ds P

DpGSq. This and (4.34) imply that v P DpGSq. The proof of Corollary 4.5.7 is thuscompleted.

4.5.5 Generators of strongly continuous semigroups

In this section we show that a strongly continuous semigroup is uniquely determinedby its generator; see Proposition 4.5.9 below. In Proposition 4.5.9 we use the assump-tion that the graph of one mapping is a subset of the graph of another mapping. Togetter a better understanding for this assumption, we first note the following remark.

Remark 4.5.8. Let A1, A2, B be sets and let f1 : A1 Ñ B and f2 : A2 Ñ B bemappings. Then it holds that Graphpf1q Ď Graphpf2q if and only if (A1 Ď A2 andf2|A1XA2 “ f1).

We are now ready to show that a strongly continuous semigroup is uniquelydetermined by its generator.

Proposition 4.5.9 (The generator determines the semigroup). Let K P tR,Cu, letpV, ¨V q be a K-Banach space, and let S, S : r0,8q Ñ LpV q be strongly continuoussemigroups with GraphpGSq Ď GraphpGSq. Then it holds that S “ S and GS “ GS.

Proof of Proposition 4.5.9. Let v P DpGSq Ď DpGSq, t P p0,8q and let η : r0, ts Ñ Vbe the function with the property that for all s P r0, ts it holds that

ηpsq “ St´s Ss v. (4.35)


Then it holds for all s P r0, ts, u P r0, ts with s ‰ u that

›

›

›

›

ηpuq ´ ηpsq

u´ s

›

›

›

›

V

“

›

›

›

›

›

Stú Su v ´ St´s Ss v

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

Stú

„

Su v ´ Ss v

u´ s

`

“

Stú ´ St´s‰

Ssv

u´ s

›

›

›

›

›

V

“

›

›

›

›

›

St´s

„

Su v ´ Ss v

u´ s

`

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

ď

›

›

›

›

›

St´s

„

Su v ´ Ss v

u´ s

´

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

›

›

›

›

›

V

`

›

›

›

›

”

Stú ´ St´s

ı

„

Su v ´ Ss v

u´ s

›

›

›

›

V

.

(4.36)

This implies that for all s P r0, ts, punqnPN Ď r0, tsztsu with limnÑ8 un “ s and alln P N it holds that

›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď

›

›

›

›

›

St´s

„

Sunv ´ Ss v

un ´ s

´

“

Stún ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` sup!

›

›Stúnw ´ St´sw›

›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(4.37)

Lemma 4.5.5 and Lemma 4.3.2 prove that for all s P r0, ts it holds that

limuÑs

«

“

Stú ´ St´s‰

Ssv

pt´ uq ´ pt´ sq

ff

“ GSSt´sSsv “ St´sGSSsv “ St´sGSSsv (4.38)

and

limuÑs

„

Suv ´ Ss v

u´ s

“ GSSsv. (4.39)

Putting (4.38)–(4.39) into (4.37) proves that for all s P r0, ts and all punqnPN Ď


r0, tsztsu with limnÑ8 un “ s it holds that

limnÑ8

›

›

›

›

ηpunq ´ ηpsq

un ´ s

›

›

›

›

V

ď limnÑ8

›

›

›

›

›

St´s

„

Sunv ´ Ss v

un ´ s

´

“

St´un ´ St´s‰

Ssv

pt´ unq ´ pt´ sq

›

›

›

›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“

›

›

›St´sGSSsv ´ St´sGSSsv

›

›

›

V

` lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

“ lim supnÑ8

sup!

›


›

V: w P tGSSsvu Y

!

Sumv´Ssvum´s

: m P N

))

.

(4.40)

This together with Lemma 4.5.5 and Lemma 4.5.4 proves that η is differentiable andthat for all s P r0, ts it holds that η1psq “ 0. This implies that

Stv “ ηp0q “ ηptq “ Stv. (4.41)

As v P DpGSq was arbitrary, we obtain that St|DpGSq “ St|DpGSq. Corollary 4.5.7 hence

proves that St “ St. This completes the proof of Proposition 4.5.9.

4.5.6 A generalization of matrix exponentials to infinite di-mensions

Proposition 4.5.9 and Definition 4.3.3 ensure that the next definition, Definition 4.5.10,makes sense.

Definition 4.5.10 (Generalized matrix exponential). LetK P tR,Cu, let pV, ¨V q bea K-Banach space, and let A : DpAq Ď V Ñ V be a generator of a strongly continuoussemigroup. Then we denote by eAt P LpV q, t P r0,8q, the linear operators with theproperty that for all strongly continuous semigroups S : r0,8q Ñ LpV q with GS “ Aand all t P r0,8q it holds that

eAt “ St. (4.42)


4.5.7 A characterization of strongly continuous semigroups

Lemma 4.5.11 (Characterization of strongly continuous semigroups). Let pV, ¨V qbe a Banach space. A semigroup S : r0,8q Ñ LpV q is strongly continuous if andonly if for all v P V it holds that limtŒ0 Stv “ v.

Proof of Lemma 4.5.11. A strongly continuous semigroup S : r0,8q Ñ LpV q clearlysatisfies that for all v P V it holds that limtŒ0 Stv “ v. In the following we thusassume that S : r0,8q Ñ LpV q is a semigroup which fulfills for all v P V thatlimtŒ0 Stv “ v. Corollary 4.5.3 hence implies that for all t P r0, T s it holds that

limsÑtSsv ´ StvV “ lim

sÑt

›

›Sminps,tq

`

S|t´s|v ´ v˘›

›

V

ď limsÑt

”

›

›Sminps,tq

›

›

LpV q

›

›S|t´s|v ´ v›

›

V

ı

ď

«

supuPr0,t`1s

SuLpV q

ff

”

limsÑt

›

›S|t´s|v ´ v›

›

V

ı

“ 0.

(4.43)


4.6 Uniformly continuous semigroups

Lemma 4.6.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup with the property that

limtŒ0St ´ S0LpV q “ 0. (4.44)

Then it holds for all t P r0,8q that supsPr0,ts SsLpV q ă 8.

Proof of Lemma 4.6.1. The assumption limtŒ0 St ´ S0LpV q “ 0 ensures that there

exists a real number ε P p0,8q such that

supsPr0,εs

SsLpV q ă 8. (4.45)

Combining this with Proposition 4.4.1 completes the proof of Lemma 4.6.1.

Lemma 4.6.2. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup. Then S is uniformly continuous if and only iflimtŒ0 St ´ S0LpV q “ 0.

4.6. UNIFORMLY CONTINUOUS SEMIGROUPS 99

Proof of Lemma 4.6.2. Clearly, it holds that if S is uniformly continuous, then itholds that limtŒ0 St ´ S0LpV q “ 0. It thus remains to prove that the condition

limtŒ0 St ´ S0LpV q “ 0 ensures that S is uniformly continuous. We thus assume inthe following that

limtŒ0St ´ S0LpV q “ 0. (4.46)

Lemma 4.6.1 hence implies that for all t P r0,8q it holds that

supsPr0,ts

SsLpV q ă 8. (4.47)

This and (4.46) show that for all t P r0,8q it holds that

limsÑtSs ´ StLpV q “ lim

sÑt

›

›Sminps,tq

“

Srmaxps,tq´minps,tqs ´ S0

‰›

›

LpV q

ď

”

limsÑt

›

›Srmaxps,tq´minps,tqs ´ S0

›

›

LpV q

ı

«

supsPr0,t`1s

SsLpV q

ff

“ 0.(4.48)


4.6.1 Matrix exponential in Banach spaces

Lemma 4.6.3. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, let A PLpV q, and let t P r0,8q. Then

8ÿ

n“0

›

›

›

›

pAtqn

n!

›

›

›

›

LpV q

ď

8ÿ

n“0

tn AnLpV qn!

“ etALpV q ă 8. (4.49)

The statement of Lemma 4.6.3 is clear. The next result, Lemma 4.6.4, demon-strates one way how uniformly continuous semigroup can be constructed. Observethat Lemma 4.6.3 ensures that the function S in Lemma 4.6.4 does indeed exist.

Lemma 4.6.4 (Matrix exponential in Banach spaces). Let K P tR,Cu, let pV, ¨V qbe a K-Banach space, let A P LpV q, and let S : r0,8q Ñ LpV q be the function withthe property that for all t P r0,8q it holds that

St “8ÿ

n“0

pAtqn

n!. (4.50)

Then it holds that S is a uniformly continuous semigroup, it holds that GS “ A, andit holds for all t P r0,8q that

StLpV q ď et GSLpV q . (4.51)


Proof of Lemma 4.6.4. First, we note that for all t1, t2 P r0,8q it holds that

St1St2 “

«

8ÿ

n“0

pAt1qn

n!

ff«

8ÿ

n“0

pAt2qn

n!

ff

“

8ÿ

n,m“0

An`m pt1qnpt2q

m

n!m!

“

8ÿ

k“0

ÿ

n,mPN0n`m“k

Ak pt1qnpt2q

m

n!m!“

8ÿ

k“0

Ak

k!

«

kÿ

n“0

k!

n! pk ´ nq!¨ pt1q

n¨ pt2q

pk´nq

ff

“

8ÿ

k“0

Ak

k!rt1 ` t2s

k“ St1`t2 .

(4.52)

This shows that S is a semigroup. Moreover, observe that for all t P r0,8q it holdsthat

St ´ S0LpV q “

›

›

›

›

›

8ÿ

n“1

pAtqn

n!

›

›

›

›

›

LpV q

ď t ALpV q

›

›

›

›

›

8ÿ

n“1

pAtqpn´1q

n!

›

›

›

›

›

LpV q

“ t ALpV q

›

›

›

›

›

8ÿ

n“0

pAtqn

pn` 1q!

›

›

›

›

›

LpV q

ď t ALpV q

«

8ÿ

n“0

AtnLpV qpn` 1q!

ff

ď t ALpV q etALpV q .

(4.53)

This proves that S is uniformly continuous. Furthermore, note that for all t P p0,8qit holds that

›

›

›

›

St ´ S0

t´ A

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

pAtqpn´1q

n!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“0

pAtqn

pn` 1q!

ff

´ A

›

›

›

›

›

LpV q

“

›

›

›

›

›

A

«

8ÿ

n“1

pAtqn

pn` 1q!

ff›

›

›

›

›

LpV q

“ t

›

›

›

›

›

A2

«

8ÿ

n“0

pAtqn

pn` 2q!

ff›

›

›

›

›

LpV q

ď t A2LpV q e

tALpV q .

(4.54)

This proves that GS “ A. Combining this with (4.49) completes the proof ofLemma 4.6.4.


4.6.2 Continuous invertibility of bounded linear operatorsin Banach spaces

Lemma 4.6.5 (Geometric series in Banach spaces and inversion of bounded linearoperators). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A P LpV q bea bounded linear operator with IdV ÁLpV q ă 1. Then it holds that A is bijective,

it holds that A´1 P LpV q, it holds thatř8

n“0 rIdV ÁsnLpV q ă 8, and it holds that

A´1“

8ÿ

n“0

rIdV Ásn . (4.55)

Proof of Lemma 4.6.5. Throughout this proof let Q P LpV q and Sn P LpV q, n P N0,be the bounded linear operators with the property that for all n P N it holds that

Q “ IdV Á and Sn “nÿ

k“0

Qk. (4.56)

Note that the assumption that QLpV q ă 1 ensures that

8ÿ

k“0

›

›Qk›

›

LpV qď

8ÿ

k“0

QkLpV q “1

“

1´ QLpV q‰ ă 8. (4.57)

This implies that Sn, n P N0, is a Cauchy-sequence in LpV q and thus convergence inLpV q. Next we claim that for all n P N0 it holds that

ASn “ IdV ´Qn`1. (4.58)

We show (4.58) by induction on n P N0. For this observe that

AS0 “ A IdV “ A “ IdV ´Q. (4.59)

This proves the base case n “ 0 in (4.58). Next note that if n P N and if ASn´1 “

IdV ´Qn, then it holds that

ASn “ A

«

nÿ

k“0

Qk

ff

“ A

«

IdV `nÿ

k“1

Qk

ff

“ A

«

IdV `Q

«

n´1ÿ

k“0

Qk

ffff

“ A rIdV `QSn´1s “ A`QASn´1 “ A`Q rIdV ´Qns

“ A`Q´Qn`1“ IdV ´Q

n`1.

(4.60)


This proves (4.58). Next note that (4.58) implies that for all n P N0 it holds that

ASn “ SnA “ IdV ´Qn`1. (4.61)

This and the fact that pSnqnPN0 Ď LpV q converges shows that

A”

limnÑ8

Sn

ı

loooomoooon

PLpV q

“

”

limnÑ8

Sn

ı

loooomoooon

PLpV q

A “ IdV . (4.62)

This implies that A is bijective and that A´1 “ limnÑ8 Sn P LpV q. The proof ofLemma 4.6.5 is thus completed.

4.6.3 Generators of uniformly continuous semigroup

Lemma 4.6.6 (The generator of a uniformly continuous semigroup). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then GS P LpV q.

Proof of Lemma 4.6.6. The assumption that S is uniformly continuous implies thatfor all t P r0,8q it holds that

limsŒ0

›

›

›

›

1

s

ż t`s

t

Su du´ St

›

›

›

›

LpV q

“ limsŒ0

›

›

›

›

1

s

ż t`s

t

rSu ´ Sts du

›

›

›

›

LpV q

ď limsŒ0

„

1

s

ż t`s

t

Su ´ StLpV q du

ď limsŒ0

«

supuPrt,t`ss

Su ´ StLpV q

ff

ď StLpV q

«

limsŒ0

«

supuPr0,ss

Su ´ S0LpV q

ffff

“ StLpV q

„

lim supsŒ0

Ss ´ S0LpV q

“ StLpV q

„

limsŒ0

Ss ´ S0LpV q

“ 0.

(4.63)

This implies that there exists a real number ε P p0,8q such that›

›

›

›

1

ε

ż ε

0

Ss ds´ S0

›

›

›

›

LpV q

ă 1. (4.64)

Lemma 4.6.5 hence shows thatşε

0Ss ds P LpV q is bijective with

„ż ε

0

Ss ds

´1

P LpV q. (4.65)


Hence, we obtain that for all t P p0, εq it holds that

St ´ IdVt

“

„

St ´ S0

t

„ż ε

0

Ss ds

„ż ε

0

Ss ds

´1

“

„

şε

0rSt`s ´ Sss ds

t

„ż ε

0

Ss ds

´1

“

«

şt`ε

tSs ds´

şε

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

“

«

şε`t

εSs ds´

şt

0Ss ds

t

ff

„ż ε

0

Ss ds

´1

.

(4.66)

This together with the identity (4.63) shows that

limtŒ0

›

›

›

›

›

St ´ IdVt

´ rSε ´ S0s

„ż ε

0

Ss ds

´1›

›

›

›

›

LpV q

“ 0. (4.67)

This proves that GS P LpV q and that

GS “ rSε ´ S0s

„ż ε

0

Ss ds

´1

. (4.68)


4.6.4 A characterization result for uniformly continuous semi-groups

Theorem 4.6.7 (Characterization of uniformly continuous semigroups). Let K P

tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup.Then the following statements are equivalent:

(i) It holds that S is uniformly continuous.

(ii) It holds that limtŒ0 St ´ S0LpV q “ 0.

(iii) It holds that GS P LpV q.


Proof of Theorem 4.6.7. Lemma 4.6.2 implies that (i) and (ii) are equivalent. More-over, Lemma 4.6.6 ensures that (i) implies (iii). It thus remains to prove that (iii)implies (i). To show this we assume for the rest of this proof that GS P LpV q. Thenlet S : r0,8q Ñ LpV q be the function with the property that for all t P r0,8q it holdsthat

St “8ÿ

n“0

ptGSqn

n!. (4.69)

Observe that Lemma 4.6.4 shows that S is uniformly continuous and that

GS “ GS. (4.70)

In the next step we apply Proposition 4.5.9 to obtain that S “ S. This proves thatS is uniformly continuous. The proof of Theorem 4.6.7 is thus completed.

4.6.5 An a priori bound for uniformly continuous semigroups

Combining Lemma 4.6.4 and Theorem 4.6.7 immediately results in the followingestimate.

Proposition 4.6.8 (A priori bounds for uniformly continuous semigroups). Let K PtR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then it holds for all t P r0,8q that

supsPr0,ts

SsLpV q ď et GSLpV q ă 8. (4.71)

4.7. SEMIGROUPS GENERATED BY DIAGONAL OPERATORS 105

4.7 Semigroups generated by diagonal operators

Theorem 4.7.1 (Semigroups generated by diagonal operators). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let B Ď H be an orthonormal basis, let λ : BÑK be a function with the property that supbPBRepλbq ă 8, and let A : DpAq Ď H Ñ

H be a linear operator with the property that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉|2 ă 8

+

(4.72)

and with the property that for all v P DpAq it holds that Av “ř

bPB λb 〈b, v〉H b.Then it holds that A is a generator of a strongly continuous semigroup, it holds forall v P H, t P r0,8q that

eAtv “ÿ

bPB

eλbt 〈b, v〉H b, (4.73)

and it holds for all t P r0,8q that eAt P LpHq is a diagonal linear operator.

Proof of Theorem 4.7.1. Let S : r0,8q Ñ LpHq be the function with the propertythat for all v P H, t P r0,8q it holds that

Stpvq “ÿ

bPB

eλbt 〈b, v〉H b. (4.74)

Note that the assumption that supbPBRepλbq ă 8 ensures that such a function doesindeed exist. Next observe that for all t1, t2 P r0,8q, v P H it holds that

St1pSt2pvqq “ St1

˜

ÿ

bPB

eλbt2 〈b, v〉H b

¸

“ÿ

bPB

eλbt2 〈b, v〉H St1pbq

“ÿ

bPB

eλbt2 〈b, v〉H

«

ÿ

cPB

eλct1 〈c, b〉H c

ff

“ÿ

bPB

eλbt2 〈b, v〉H“

eλbt1b‰

“ÿ

bPB

eλbpt1`t2q 〈b, v〉H b “ St1`t2pvq.

(4.75)

The function S is thus a semigroup. Moreover, observe that Lebesgue’s theorem of


dominated convergence proves that for all v P H it holds that

limtŒ0Stv ´ v

2H “ lim

tŒ0

›

›

›

›

›

ÿ

bPB

“

eλbt ´ 1‰

〈b, v〉H b

›

›

›

›

›

2

H

“ limtŒ0

«

ÿ

bPB

›

›

“

eλbt ´ 1‰

〈b, v〉H b›

›

2

H

ff

“ limtŒ0

«

ÿ

bPB

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2

ff

“ limtŒ0

ż

B

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2 #Bpdbq

“

ż

B

limtŒ0

ˇ

ˇeλbt ´ 1ˇ

ˇ

2|〈b, v〉H |

2 #Bpdbq “ 0.

(4.76)

This completes the proof of Theorem 4.7.1.

Proposition 4.7.2 (Semigroups generated by diagonal operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then it holds that supλPσP pAqRepλq ă 8 if and only if A is agenerator of a strongly continuous semigroup.

Proof of Proposition 4.7.2. Theorem 4.7.1 shows that the condition supλPσP pAqRepλq ă8 implies that A is a generator of a strongly continuous semigroup. In remainder ofthis proof we thus assume that A is the generator of a strongly continuous semigroupS : r0,8q Ñ LpHq. Then we note that the assumption that A is diagonal ensuresthat there exists an orthonormal basis B Ď H of H and a function λ : B Ñ K suchthat

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.77)


Av “ÿ

bPB

λb 〈b, v〉H b. (4.78)

The fact that GS “ A implies that for all b P B it holds that the function

r0,8q Q t ÞÑ Stb P H (4.79)

is continuously differentiable and that for all t P r0,8q, v P H it holds that

〈v, S0pbq〉H “ 〈v, b〉H and ddt〈v, Stpbq〉H “ 〈v,GSStpbq〉H “ λb 〈v, Stpbq〉H .

(4.80)


This shows that for all b P B, v P H, t P r0,8q it holds that

〈v, Stb〉H “ eλbt 〈v, b〉H . (4.81)

Hence, we obtain that for all b P B, t P r0,8q it holds that

Stb “ eλbtb. (4.82)

This, in turn, ensures that

8 ą S1LpHq ě supbPB

S1bH “ supbPB

ˇ

ˇeλbˇ

ˇ “ supbPB

ˇ

ˇeRepλbqˇ

ˇ “ esupbPBRepλbq. (4.83)

This implies that supbPBRepλbq ă 8. The proof of Proposition 4.7.2 is thus com-pleted.

4.7.1 Semigroup generated by the Laplace operator

Example 4.7.3 (Heat equation with Dirichlet boundary conditions). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq and let v : p0, 1q Ñ R be a twice continuouslydifferentiable function with vp0`q “ vp1´q “ 0. Then it holds that supλPσP pAq λ “´π2 ă 8 and Theorem 4.7.1 hence ensures that A is the generator of a stronglycontinuous semigroup r0,8q Q t ÞÑ eAt P LpHq. Moreover, the function u : r0,8q ˆp0, 1q Ñ R with the property that

@ t P r0,8q, x P p0, 1q : upt, xq “ peAtvqpxq (4.84)

is twice continuously differentiable and satisfies that for all pt, xq P r0,8q ˆ p0, 1q itholds that

B

Btupt, xq “ B2

Bx2upt, xq, up0, xq “ vpxq, upt, 0`q “ upt, 1´q “ 0. (4.85)


Example 4.7.4 (Laplace operator on L2pBorelp0,1q; |¨|Rq without boundary condi-tions). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be a linear operatorwith DpAq “ H2pp0, 1q;Rq and with the property that for all v P DpAq it holds thatAv “ v2. Then A is not a diagonal linear operator in the sense of Definition 3.5.1 andA is not a generator of a strongly continuous semigroup. Indeed, observe that for allv P L2pBorelp0,1q; |¨|Rq with vpxq “ vpyq for all x, y P p0, 1q it holds that Av “ v2 “ 0and this shows that 0 P σP pAq and that 1 P Kernp0 ´ Aq “ KernpAq. Moreover,note that for all n P N, x P p0, 1q it holds that d2

dx2sinpnπxq “ ń2π2 sinpnπxq and

this proves that for all n P N it holds that ń2π2 P σP pAq and psinpnπxqqxPp0,1q P

Kernpń2π2 ´ Aq. Furthemore, note thatş1

01 ¨ sinpπxq dx ‰ 0 and this implies that

it does not hold that for all v P Kernp0 ´ Aq, w P Kernpń2π2 ´ Aq it holds thatş1

0vpxqwpxq dx “ 0. Proposition 3.5.13 hence proves that A is not a diagonal oper-

ator in the sense of Definition 3.5.1. Moreover, Proposition 4.5.9 implies that A isnot the generator of a strongly continuous semigroup.

4.7.2 Smoothing effect of the semigroup

Proposition 4.7.5. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric linear operator with infpσP pAqq ą 0, let B Ď Hbe an orthonormal basis, let λ : BÑ K be a function such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.86)


Av “ÿ

bPB

λb 〈b, v〉H b, (4.87)

and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated toA. Then

• it holds that B Ď pXrPRHrq,

• it holds for all r P R that spanpBqHr“ Hr, and

• it holds for all r P R that A´rpBq “

bpλbqr

P H : b P B(

is an orthonormalbasis of Hr.


Proof of Proposition 4.7.5. Observe that Proposition 4.7.5 follows immediately fromDefinition 3.5.16, Definition 3.5.17, and Definition 3.5.25.

In the next result, Theorem 4.7.6, we establish a smoothing effect for strongly con-tinuous semigroups generated by diagonal linear operators. We recall Remark 2.5.21for the formulation of Theorem 4.7.6.

Theorem 4.7.6 (Smoothing effect of semigroups generated by diagonal operators).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ Hbe a symmetric linear operator with suppσP pAqq ă 0, let B Ď H be an orthonormalbasis, let λ : BÑ K be a function such that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(4.88)


Av “ÿ

bPB

λb 〈b, v〉H b, (4.89)

and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to´A. Then

(i) it holds that for all r P r0,8q that

suptPr0,8q

›

›p´tAqreAt›

›

LpHqď

”r

e

ır

ă 8, (4.90)

(ii) it holds for all t P p0,8q, r P p´8, 0q, v P H that eAtvH ď |r||r| tr vHr ă 8,

(iii) it holds for all t P p0,8q, r P R that

eAtpHrq Ď pXsPRHsq (4.91)

(cf. Item (ii) and Proposition 2.5.19),

(iv) it holds for all t P r0,8q, r P p´8, 0q, v P H that etAvHr ď vHr , and

(v) it holds for all t P r0,8q, v P pYrPRHrq that

eAtv “ÿ

bPB

eλbt 〈b, v〉H b (4.92)

(cf. Item (iv) and Proposition 2.5.19).


Proof of Theorem 4.7.6. Observe that Proposition 4.7.5 implies that for all r P r0,8qit holds that

›

›p´tAqreAt›

›

LpHq“ sup

bPB

ˇ

ˇp´tλbqreλbt

ˇ

ˇ ď supxPp0,8q

„

xr

ex

ď

”r

e

ır

ă 8. (4.93)


Lemma 4.7.7. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 0.Then it holds for all t P p0,8q, r P r0, 1s that

›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHqď 1. (4.94)

Proof of Lemma 4.7.7. Observe that for all t P p0,8q, r P r0, 1s it holds that

›

›p´tAq´r`

eAt ´ IdH˘›

›

LpHq“ sup

λPσP ptAqq

ˇ

ˇp´λq´r`

eλ ´ 1˘ˇ

ˇ

ď supxPp0,8q

„

p1´ e´xq

xr

ď 1.(4.95)


Exercise 4.7.8. Let T P p0,8q, r P r0, 1q, K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operator withsup

`

σP pAq˘

ă 0, and let e : r0, T s Ñ H be a continuous function with the property

that for all t P r0, T s it holds that eptq “ ep0q `şt

0p´Aqr ept´sqA epsq ds. Prove that

suptPr0,T s eptqH ď ep0qH ¨ E1´r

“

T 1´r‰

.

Part II

Foundations in probability theory

111

Chapter 5

Random variables with values ininfinite dimensional spaces

In the most of this chapter we follow the presentations in Da Prato & Zabczyk [7],Werner [29] and Prevot & Rockner [24].

5.1 Borel sigma-algebras on normed vector spaces

5.1.1 The Hahn-Banach theorem

We first recall the Hahn-Banach theorem (see, e.g., Theorem III.1.5 in Werner [29]).

Theorem 5.1.1 (Hahn-Banach theorem; Extension of continuous linear functionals).Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, let U Ď V be a K-subspaceof V , and let φ P U 1. Then there exists a ϕ P V 1 with the property that

ϕ|U “ φ and ϕV 1 “ φU 1 . (5.1)

The proof of Theorem 5.1.1 uses the axiom of choice. The next corollary is animmediate consequence of the Hahn-Banach theorem.

Corollary 5.1.2 (Projections into 1-dimensional subspaces). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space with V ‰ t0u, and let v P V . Then there existsa ϕ P V 1 with the property that

ϕpvq “ vV and ϕV 1 “ 1. (5.2)

113

114 CHAPTER 5. RANDOM VARIABLES

Proof of Corollary 5.1.2. We show Corollary 5.1.2 in two steps. In the first step weassume that v ‰ 0. Let U be the K-subspace of V given by U “ tλv P V : λ P Ku “spantvu and let φ : U Ñ K be the mapping with the property that for all λ P K itholds that

φpλvq “ λ vV . (5.3)

Theorem 5.1.1 implies the existence of a ϕ P V 1 with the property that

ϕ|U “ φ and ϕV 1 “ φU 1 “ 1. (5.4)

This proves (5.2) in the case v ‰ 0. In the second step we assume that v “ 0. Theassumption that V ‰ t0u then shows that there exists u P V with the property thatu ‰ 0. The first step hence proves that there exists a ϕ P V 1 with the property that

ϕpuq “ uV and ϕV 1 “ 1. (5.5)

In addition, observe that ϕpvq “ ϕp0q “ 0 “ vV . The proof of Corollary 5.1.2 isthus completed.

5.1.2 Norm representations in normed vector spaces

The next result, Corollary 5.1.3, is an immediate consequence of Corollary 5.1.2above.

Corollary 5.1.3 (Norm via the dual space). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space with V ‰ t0u, and let v P V . Then

vV “ supϕPV 1zt0u

ϕpvq

ϕV 1“ sup

ϕPV 1zt0u

|ϕpvq|

ϕV 1. (5.6)

If the normed vector space in Corollary 5.1.3 is separable, then the followingresult, Corollary 5.1.4, can be obtained. Corollary 5.1.4 is also an immediate conse-quence of Corollary 5.1.2 above.

Corollary 5.1.4 (Norm of a separable normed vector space via the dual space). LetK P tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then there existsa sequence pϕnqnPN Ď V 1 with the property that for all v P V it holds that

vV “ supnPN

ϕnpvq “ supnPN

|ϕnpvq| . (5.7)

5.1. BOREL SIGMA-ALGEBRAS ON NORMED VECTOR SPACES 115

Proof of Corollary 5.1.4. W.l.o.g. we assume that V ‰ t0u. The assumption thatpV, ¨V q is separable implies that there exists a sequence vn P V , n P N, with theproperty that the set tvn : n P Nu is dense in V . Corollary 5.1.2 hence shows thatthere exists a sequence ϕn P V

1, n P N, with the property that for all n P N it holdsthat

ϕnpvnq “ vnV and ϕnV 1 “ 1. (5.8)

This implies that for all k P N it holds that

vkV “ supnPN

ϕnpvkq. (5.9)

Next let v P V and ε P p0,8q be arbitrary. Then observe that

supnPN

ϕnpvq ď supnPN

rϕnV 1 vV s “ vV . (5.10)


vV ď ε` supnPN

ϕnpvq. (5.11)

To see this observe that the fact that tvk P V : k P Nu is dense in V ensures thatthere exists a k P N such that v ´ vkV ď

ε2. This implies that

vV ď vkV ` v ´ vkV “ ϕkpvkq ` v ´ vkV“ ϕkpvq ` v ´ vkV ` ϕkpvk ´ vq

ď ϕkpvq ` v ´ vkV ` ϕkV 1 v ´ vkV“ ϕkpvq ` 2 v ´ vkV ď sup

nPNϕnpvq ` 2 v ´ vkV ď sup

nPNϕnpvq ` ε.

(5.12)


5.1.3 Linear characterization of the Borel sigma-algebra

Proposition 5.1.5 (Linear characterization of the Borel sigma-algebra). Let K P

tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then there exists asequence ϕn P V

1, n P N, such that

BpV q “ σV ppϕqϕPV 1q “ σV pϕ : ϕ P V 1q “ σV pϕn : n P Nq . (5.13)


Proof of Proposition 5.1.5. Let fv : V Ñ r0,8q, v P V , be the functions with theproperty that for all x, v P V it holds that

fvpxq “ x´ vV . (5.14)

Observe thatBpV q “ σV pfv : v P V q . (5.15)

Next observe that Corollary 5.1.4 implies that there exists a sequence ϕn P V1, n P N,

such that for all v P V it holds that

vV “ supnPN

ϕnpvq. (5.16)

This implies that

σV pϕn : n P Nq “ σV`

ϕnpp¨q ` vq : n P N, v P V˘

Ě σV pfv : v P V q . (5.17)


5.2 Measures on normed vector spaces

5.2.1 Uniqueness theorem for measures

Definition 5.2.1 (Image measure/Push forward measure). Let pΩ,A, µq be a mea-sure space, let pΩ, Aq be a measurable space, and let f : Ω Ñ Ω be an A/A-measurablemapping. Then we denote by fpµq “ fpµqA : AÑ r0,8s the function with the prop-erty that for all A P A it holds that

`

fpµq˘

pAq “`

fpµqA˘

pAq “ µ`

f´1pAq

˘

(5.18)

and we call fpµq (we call fpµqA) the image measure of f .

Definition 5.2.2 (Finiteness of measures). Let pΩ,F , µq be a measure space withµpΩq ă 8. Then µ is called finite.

Definition 5.2.3 (Sigma-finiteness of measures). Let pΩ,F , µq be a measure spacewith the property that there exists a sequence An P F , n P N, of sets such thatYnPNAn “ Ω and such that for all n P N it holds that µpAnq ă 8. Then µ is calledsigma-finite.

Definition 5.2.4 (X-Stability). Let Ω be a set and let E Ď PpΩq be a set with theproperty that for all a, b P E it holds that aX b P E. Then E is called X-stable.

5.2. MEASURES ON NORMED VECTOR SPACES 117

Theorem 5.2.5 (Uniqueness theorem for measures). Let Ω be a set, let E Ď PpΩqbe a X-stable subset of PpΩq, let µk : σΩpEq Ñ r0,8s, k P t1, 2u, be measures withthe property that µ1|E “ µ2|E and with the property that there exists a non-decreasingsequence Ωn P tA P E : µ1pAq ă 8u, n P N, such that YnPNΩn “ Ω. Then µ1 “ µ2.

5.2.2 Fourier transform of a measure

5.2.2.1 Characteristic functionals

Theorem 5.2.5 is, for example, established as Lemma 1.42 in Klenke [17].

Proposition 5.2.6 (Characteristic function). Let d P N and let µk : BpRdq Ñ r0,8s,k P t1, 2u, be finite measures with the property that for all ξ P Rd it holds that

ż

Rd

ei〈ξ,x〉Rd µ1pdxq “

ż

Rd

ei〈ξ,x〉Rd µ2pdxq. (5.19)

Then µ1 “ µ2.

Proposition 5.2.6 is, for example, proved as Theorem 15.8 in Klenke [18].

Definition 5.2.7 (Characteristic functional). Let pV, ¨V q be a normed R-vectorspace. Then we denote by

FV :

µ PMpBpV q, r0,8sq : µ is a finite measure on pV,BpV qq(

ÑMpV 1,Cq(5.20)

the mapping with the property that for all µ P DpFV q, ϕ P V1 it holds that

pFV µqpϕq “`

FV pµq˘

pϕq “

ż

V

ei¨ϕpxq µpdxq (5.21)

and for every µ P DpFV q we call FV pµq the characteristic functional of µ.

Lemma 5.2.8 (Elementary properties of the characteristic functionals). Let pV, ¨V qbe a normed R-vector space. Then

• for all µ P DpFV q it holds that pFV µqp0q “ µpV q,

• for all µ1, µ2 P DpFV q, a P r0,8q it holds that FV paµ1 ` µ2q “ aFV pµ1q `

FV pµ2q, and

• impFV q Ď CpV 1,Cq.


Proof of Lemma 5.2.8. First of all, observe that for all µ P DpFV q it holds that

pFV µqp0q “

ż

V

ei0 µpdxq “

ż

V

1µpdxq “ µpV q. (5.22)

Next note that for all µ1, µ2 P DpFV q, a P r0,8q, ϕ P V1 it holds that

`

FV paµ1 ` µ2q˘

pϕq “

ż

V

eiϕpxq ra ¨ µ1pdxq ` µ2pdxqs

“ a

ż

V

eiϕpxqµ1pdxq `

ż

V

eiϕpxqµ2pdxq

“ aFV pµ1q ` FV pµ2q.

(5.23)

Finally, observe that Lebesgue’s theorem of dominated convergence proves that forall µ P DpFV q and all ϕn P V

1, n P N0, with limnÑ8 ϕn ´ ϕ0V 1 “ 0 it holds that

limnÑ8

|pFV µqpϕnq ´ pFV µqpϕ0q| ď limnÑ8

ż

V

ˇ

ˇeiϕnpxq ´ eiϕ0pxqˇ

ˇµpdxq

“

ż

V

limnÑ8

ˇ

ˇeiϕnpxq ´ eiϕ0pxqˇ

ˇµpdxq “ 0.

(5.24)


5.2.2.2 Fourier transform on separable normed vector spaces

Lemma 5.2.9 (Characteristic functional determines measure uniquely). Let pV, ¨V qbe a separable normed R-vector space. Then FV is injective.

Proof of Lemma 5.2.9. Let µ1, µ2 P DpFV q satisfy FV pµ1q “ FV pµ2q. Then notethat for all n P N, φ “ pφ1, . . . , φnq P LpV,R

nq, ξ P Rn it holds thatż

Rn

ei〈ξ,x〉Rn`

φpµ1q˘

pdxq “

ż

V

ei〈ξ,φpvq〉Rn pµ1qpdvq “`

FV µ1

˘

´

xξ, φp¨qyRn¯

“`

FV µ2

˘

´

xξ, φp¨qyRn¯

“

ż

V

ei〈ξ,φpvq〉Rn pµ2qpdvq “

ż

Rn

ei〈ξ,x〉Rn`

φpµ2q˘

pdxq.

(5.25)

Proposition 5.2.6 hence implies that for all n P N, φ P LpV,Rnq it holds that

φpµ1q “ φpµ2q. (5.26)

In the next step let E Ď PpV q be the set given by

E “ď

nPN

φ´1pBq P PpV q : φ P LpV,Rn

q, B P BpRnq(

. (5.27)


Note that E Ď BpV q and observe that (5.26) shows that

µ1|E “ µ2|E . (5.28)

This, the fact that E is X-stable, the fact V P E , and Theorem 5.2.5 imply that

µ1|σV pEq “ µ2|σV pEq. (5.29)

Moreover, observe that Proposition 5.1.5 proves that

σV pEq “ BpV q. (5.30)

Combining this with (5.29) completes the proof of Lemma 5.2.9.

5.2.2.3 Almost surely separably supported

In Theorem 5.2.30 below we prove a generalization of Lemma 5.2.9. For this weneed a few preparations. These preparations and Theorem 5.2.30 are based on thepresentations in Van Neerven [27].

Lemma 5.2.10. Let pE, Eq be a topological space and let A Ď E be separable. ThenA is separable too.

Lemma 5.2.11. Let pE, Eq be a topological space and let A Ď E and B Ď E beseparable. Then AYB is separable too.

Lemma 5.2.12. Let pV, ¨V q be a normed vector space and let A Ď V be separable.Then spanpAq is separable too.

Definition 5.2.13 (Support of a measure). Let pE, Eq be a topological space and letµ : BpEq Ñ r0,8s be a measure on pE,BpEqq. Then we denote by supppµq the setgiven by

supppµq “ tx P E : p@U P E : x P U ñ µpUq ą 0qu (5.31)

and we call supppµq the support of µ.

Question 5.2.14. Let x P R. What is supp`

δRx |BpRq˘

?

Question 5.2.15. Let d P N. What is supppλRdq?

Exercise 5.2.16. Let pE, Eq be a topological space and let µ : BpEq Ñ r0,8s be ameasure on pE,BpEqq. Prove then that supppµq is a closed set in pE, Eq, i.e., provethat Ez supppµq P E.


Remark 5.2.17. In general it is not true that µ`

Ez supppµq˘

“ 0. (see Wikipedia:support (measure theory)).

Definition 5.2.18 (Almost surely separably supported). Let pE, Eq be a topologicalspace and let µ : BpEq Ñ r0,8s be a measure with the property that there exists aseparable and closed subset A Ď E of E such that µpEzAq “ 0. Then µ is called a.s.separably supported (almost surely separably supported).

5.2.2.4 Trace set

Let pΩ,Fq be a measurable space (i.e., Ω is a set and F is a sigma-algebra on Ω)and let A Ď Ω be a subset of Ω. In some situations we are interested to have anappropriate measurable structure (an appropriate sigma-algebra) on A too. Thetrace set in the following definition provides an appropriate concept for this issue;see Lemma 5.2.20 below.

Definition 5.2.19 (Trace set). Let A and Ω be sets and let A Ď PpΩq be a subsetof the power set of Ω. Then we denote by A \A the set given by

A \A “ tAXB P PpAq : B P Au (5.32)

and we call A \A the trace set (of A in A).

Lemma 5.2.20 (Trace sigma-algebra). Let pΩ,Aq be a measurable space and letA Ď Ω be a subset of Ω (which is not necessarily an element of A). Then it holdsthat pA,A \Aq is a measurable space.

The proof of Lemma 5.2.20 is clear and therefore omitted. The next lemma andits proof can, e.g., be found as Corollary 1.83 in Klenke [18].

Lemma 5.2.21 (Trace sigma-algebras and generation of sigma-algebras). Let Ω bea set, let A Ď PpΩq be a subset of the power set of Ω, and let A Ď Ω be a subset ofΩ. Then

A \ σΩpAq “ σApA \Aq . (5.33)

Proof of Lemma 5.2.21. Let ι : A Ñ Ω be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that

ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.34)


This implies that

σApA \Aq “ σAptAXB P PpAq : B P Auq “ σA`

ι´1pBq P PpAq : B P A

(˘

“

ι´1pBq P PpAq : B P σΩpAq

(

“ tAXB P PpAq : B P σΩpAqu “ A \ σΩpAq .(5.35)


Lemma 5.2.22 (Trace topology). Let pE, Eq be a topological space and let A Ď Ebe a subset of A. Then it holds that pA,A \ Eq is a topological space.


Definition 5.2.23 (Generation of topologys). Let E be a set and let E Ď PpEq bea subset of the power set of E. Then we denote by τEpEq the set given by

τEpEq “č

A is a topologyon E with EĎA

A (5.36)

and we call τEpEq the smallest topology on E which contains E.

Lemma 5.2.24 (Topological spaces and continuous mappings). Let E and F be sets,let F Ď PpF q be a subset of the power set of F , and let f : E Ñ F be a mapping.Then

τE`

f´1pAq : A P F

(˘

“

f´1pAq : A P τF pFq

(

. (5.37)

Proof of Lemma 5.2.24. Throughout this proof let E Ď PpEq, E Ď PpEq and F ĎPpF q be the sets given by

E “ τE`

f´1pAq : A P F

(˘

, E “


(

(5.38)

and F “

A P PpF q : f´1pAq P E

(

. (5.39)

Observe that pE, Eq, pE, Eq and pF, Fq are topological spaces and that F Ď F andtf´1pAq : A P Fu Ď E . Hence, we obtain that

τF pFq Ď F and E Ď E . (5.40)

This proves that

E “


(

Ď

f´1pAq : A P F

(

Ď E Ď E . (5.41)

This shows that E “ E . The proof of Lemma 5.2.24 is thus completed.


Lemma 5.2.25 (Trace topologys and generation of topologys). Let E be a set, letE Ď PpEq be a subset of the power set of E, and let A Ď E be a subset of E. Then

A \ τEpEq “ τApA \ Eq . (5.42)

Proof of Lemma 5.2.25. Let ι : A Ñ E be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that

ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.43)

This and Lemma 5.2.24 imply that

τApA \ Eq “ τAptAXB P PpAq : B P Euq “ τA`

ι´1pBq : B P E

(˘

“

ι´1pBq : B P τEpEq

(

“ tAXB P PpAq : B P τEpEqu “ A \ τEpEq .(5.44)


Lemma 5.2.26 (Open balls generate the topologys associated to a distance-typefunction). Let E be a set, let T Ď R be a set, and let d : E ˆ E Ñ T be a functionwith the property that @x, y, z P E : dpx, xq ď 0 and dpx, zq ď dpx, yq ` dpy, zq. Then

τpdq “ τE`

ty P E : dpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

. (5.45)

Lemma 5.2.26 is an immediate consequence from Defintion 2.5.3 and Lemma 2.5.4.In the next result, Corollary 5.2.27, we study the trace set of a topological space as-sociated to a metric.

Corollary 5.2.27 (Traces and metric spaces). Let pE, dEq be a metric space and letA Ď E be a subset of E. Then A \ τpdEq “ τpdE|AˆAq.

Proof of Corollary 5.2.27. Lemma 5.2.25 and Lemma 5.2.26 imply that

A \ τpdEq “ A \ τE`

ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

“ τA`

A \

ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘

“ τA`

ty P A : dEpx, yq ă εu P PpAq : x P E, ε P p0,8q(˘

“ τA`

ty P A : dEpx, yq ă εu P PpAq : x P A, ε P p0,8q(˘

“ τpdE|AˆAq.

(5.46)



Corollary 5.2.28 (Traces and Borel sigma algebras on topological spaces). LetpE, Eq be a topological space and let A Ď E be a subset of E. Then

A \ BpEq “ A \ σEpEq “ σApA \ Eq “ BpAq. (5.47)

Corollary 5.2.28 is an immediate consequence from Lemma 5.2.21. The nextresult, Corollary 5.2.29, specalises Corollary 5.2.28 to the case where the underlyingtopological space is generated by a metric.

Corollary 5.2.29 (Traces and Borel sigma algebras on metric spaces). Let pE, dEqbe a metric space and let A Ď E be a subset of E. Then

A \ BpEq “ A \ σE`

τpdEq˘

“ σA`

τpdE|AˆAq˘

“ BpAq. (5.48)

Corollary 5.2.29 is an immediate consequence of Corollary 5.2.28 and Corol-lary 5.2.27.

5.2.2.5 Fourier transform on normed vector spaces

In Lemma 5.2.9 it has been proved that the Fourier transforms of measures on sepa-rable normed vector spaces determine the measures uniquely. The next result, The-orem 5.2.30, provides a generalization to Lemma 5.2.9. The proof of Theorem 5.2.30uses Corollary 5.2.29 above.

Theorem 5.2.30 (Characteristic functional). Let pV, ¨V q be a normed R-vectorspace. Then FV |tµPDpFV q : µ is a.s. separably supportedu is injective.

Proof of Theorem 5.2.30. Let µ1, µ2 P DpFV q be two a.s. separably supported finitemeasures with the property that

FV pµ1q “ FV pµ2q (5.49)

The assumption that µ1 and µ2 are a.s. separably supported ensures that there existseparable and closed sets A1, A2 P BpV q with the property that

µ1pV zA1q “ µ2pV zA2q “ 0. (5.50)

Lemma 5.2.11 implies that the set A1 Y A2 is separable. This and Lemma 5.2.12prove that the set

spanpA1 Y A2q (5.51)


is separable. Next let U be the set given by

U “ spanpA1 Y A2q. (5.52)

Lemma 5.2.10 and (5.50) prove that U is separable. The pair pU, ¨V |Uq is thus aclosed and separable R-vector subspace of pV, ¨V q. Moreover, equation (5.50) andthe fact that A1 Ď U and A2 Ď U prove that

µ1pV zUq “ µ2pV zUq “ 0. (5.53)

Next let ϕ P U 1 “ LpU,Rq be arbitrary. Theorem 5.1.1 then implies that there existsa ψ P V 1 “ LpV,Rq with the property that ψ|U “ ϕ. Equation (5.49) hence provesthat

`

FV µ1

˘

pψq “`

FV µ2

˘

pψq. (5.54)

Next note that Corollary 5.2.29 and the fact that U P BpV q imply that BpUq “U \BpV q “ PpUqXBpV q. This and (5.53) ensure that for all k P t1, 2u it holds that

`

FV µk˘

pψq “

ż

V

ei¨ψpxq µkpdxq “

ż

V

1Upxq ei¨ψpxq µkpdxq

“

ż

V

1Upxq ei¨ϕpxq µkpdxq “

ż

U

1Upxq ei¨ϕpxq µk|BpUqpdxq “

`

FU µk|BpUq˘

pϕq.

(5.55)

Combining (5.54) and (5.55) proves that`

FU µ1|BpUq˘

pϕq “`

FU µ2|BpUq˘

pϕq. Asϕ P U 1 was arbitrary, we obtain that

FU`

µ1|BpUq˘

“ FU`

µ2|BpUq˘

. (5.56)

Lemma 5.2.9 and the fact that U is separable hence imply that µ1|BpUq “ µ2|BpUq.This, (5.53) and the fact that BpUq “ U \ BpV q (see Corollary 5.2.29 above) implythat for all A P BpV q it holds that

µ1pAq “ µ1pAX Uq ` µ1pAzUq “ µ1pAX Uq “ µ1|U\BpV qpAX Uq

“ µ1|BpUqpAX Uq “ µ2|BpUqpAX Uq “ µ2pAX Uq

“ µ2pAzUq ` µ2pAX Uq “ µ2pAq.

(5.57)


Question 5.2.31. Let pV, ¨V q be a normed R-vector space and let µ : BpV q Ñ r0,8sbe a finite measure on pV,BpV qq. What is then the characteristic functional FV pµq?

Question 5.2.32. Let pV, ¨V q be a normed R-vector space and let µ1, µ2 : BpV q Ñr0,8s be two finite measures on pV,BpV qq with FV pµ1q “ FV pµ2q. Provide a condi-tion which is sufficient to ensure that µ1 “ µ2.


5.2.3 Covariance of a measure

5.2.3.1 The Baire category theorem on complete metric spaces

Lemma 5.2.33 (A set contains an open ball). Let pE, dEq be a metric space andlet A Ď E. Then Ac ‰ E if and only if there exist ε P p0,8q, x P E such thatty P E : dEpx, yq ă εu Ď A.

Proof of Lemma 5.2.33. Observe that

Ac “ E

ô Ac is dense in E

ô @x P E : @ ε P p0,8q : D y P Ac : dEpx, yq ă ε

ô @x P E : @ ε P p0,8q : Ac X ty P E : dEpx, yq ă εu ‰ H.

(5.58)

This implies that

Ac ‰ E

ô Dx P E : D ε P p0,8q : Ac X ty P E : dEpx, yq ă εu “ H

ô Dx P E : D ε P p0,8q : ty P E : dEpx, yq ă εu Ď A.

(5.59)


Theorem 5.2.34 (Baire category theorem for complete metric spaces). Let pE, dEqbe a complete metric space and let An Ď E, n P N, be a sequence of closed subsetsof E with the property that rYnPNAns

c‰ E. Then there exists an N P N such that

rAN sc‰ E.

5.2.3.2 Regularities for correlations on normed vector spaces

Proposition 5.2.35 (A boundedness result for correlations on normed vector spaces).Let K P tR,Cu, r P p0,8q, let pV, ¨V q be a normed K-vector space, and letµ : BpV q Ñ r0,8s be a measure with the property that for all ϕ P V 1 it holds thatş

V|ϕpxq|r µpdxq ă 8. Then

supϕPV 1zt0u

„

ş

V|ϕpxq|r µpdxq

ϕrV 1

ă 8. (5.60)


Proof of Proposition 5.2.35. Throughout this proof let Vn Ď V 1, n P N, be the setswith the property that for all n P N it holds that

Vn “"

ϕ P V 1 :

ż

V

|ϕpxq|r µpdxq ď n

*

. (5.61)

Fatou’s lemma proves that for all n P N, ψ P V 1, pϕkqkPN Ď Vn with limkÑ8 ϕk ´ ψV 1 “0 it holds that

ż

V

|ψpxq|r µpdxq “

ż

V

ˇ

ˇ

ˇlimkÑ8

ϕkpxqˇ

ˇ

ˇ

r

µpdxq “

ż

V

limkÑ8

|ϕkpxq|r µpdxq

“

ż

V

lim infkÑ8

|ϕkpxq|r µpdxq ď lim inf

kÑ8

ż

V

|ϕkpxq|r µpdxq ď lim inf

kÑ8rns “ n.

(5.62)

This implies that for every n P N it holds that Vn Ď V 1 is a closed subset of V 1. Theassumption that for all ϕ P V 1 it holds that

ş

V|ϕpxq|r µpdxq ă 8 proves that

YnPNVn “ V 1. (5.63)

The fact that Vn Ď V 1, n P N, are closed sets, the fact that pV 1, ¨V 1q is complete (seeLemma 3.4.10) and the Baire category theorem (see Theorem 5.2.34) hence prove

that there exists an N P N such that rVN sc ‰ V 1. Lemma 5.2.33 therefore showsthat there exist ψ P VN , ε P p0,8q such that

tϕ P V 1 : ϕ´ ψV 1 ď εu Ď VN . (5.64)

This implies that for all ϕ P V 1 with ϕV 1 ď ε it holds that

ż

V

|ϕpxq|r µpdxq “

ż

V

|pϕ` ψqpxq ´ ψpxq|r µpdxq

ď 2r„ż

V

|pϕ` ψqpxq|r µpdxq `

ż

V

|ψpxq|r µpdxq

ď 2r„

N `

ż

V

|ψpxq|r µpdxq

ď 2pr`1qN ă 8.

(5.65)

This, in turn, proves that for all ϕ P V 1zt0u it holds that

ż

V

|ϕpxq|r µpdxq “ϕrV 1

εr

ż

V

ˇ

ˇ

ˇ

ˇ

ε ¨ ϕpxq

ϕV 1

ˇ

ˇ

ˇ

ˇ

r

µpdxq ď2pr`1qN ϕrV 1

εră 8. (5.66)

This implies (5.60). The proof of Proposition 5.2.35 is thus completed.


The next result, Corollary 5.2.36, specialises Proposition 5.2.35 to the case wherer P p0,8q in Proposition 5.2.35 is a natural number.

Corollary 5.2.36 (A continuity result for correlations on normed vector spaces).Let K P tR,Cu, k P N, let pV, ¨V q be a normed K-vector space, and let µ : BpV q Ñr0,8s be a measure with the property that for all ϕ P V 1 it holds that

ş

V|ϕpxq|k µpdxq ă

8. Then it holds that

supϕ1,...,ϕkPV 1zt0u

„

ş

V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq

ϕ1V 1 ¨ . . . ¨ ϕkV 1

ă 8 (5.67)

and it holds that the symmetric k-linear form

V 1 ˆ ¨ ¨ ¨ ˆ V 1 Q pϕ1, . . . , ϕkq ÞÑ

ż

V

ϕ1pxq . . . ϕkpxqµpdxq P K (5.68)

is continuous.

Proof of Corollary 5.2.36. Proposition 5.2.35 implies that

supϕPV 1zt0u

«

ş

V|ϕpxq|k µpdxq

ϕkV 1

ff

ă 8. (5.69)

Holder’s inequality hence shows that

supϕ1,...,ϕkPV 1zt0u

„

ş

V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq

ϕ1V 1 ¨ . . . ¨ ϕkV 1

ď supϕ1,...,ϕkPV 1zt0u

«

kź

l“1

«

ş

V|ϕlpxq|

k µpdxq

ϕlkV 1

ffff1k

“

«

supϕPV 1zt0u

ş

V|ϕpxq|k µpdxq

ϕkV 1

ff1k

ă 8.

(5.70)

This proves (5.67). Inequality (5.67), in turn, establishes (5.68). The proof of Corol-lary 5.2.36 is thus completed.


5.2.3.3 Covariances of measures and random variables

Definition 5.2.37. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, andlet µ : BpV q Ñ r0,8s be a probability measure with the property that for all ϕ P V 1

it holds thatş

V|ϕpvq|2 µpdvq ă 8. Then we denote by Covpµq : V 1 ˆ V 1 Ñ K the

mapping with the property that for all ϕ, ψ P V 1 it holds that

pCov µqpϕ, ψq “`

Covpµq˘

pϕ, ψq

“

ż

V

„

ϕpvq ´

ż

V

ϕpuqµpduq

„

ψpvq ´

ż

V

ψpuqµpduq

µpdvq

“

ż

V

ϕpvqψpvqµpdvq ´

„ż

V

ϕpvqµpdvq

„ż

V

ψpvqµpdvq

(5.71)

and we call Covpµq the covariance of µ.

Definition 5.2.38 (Covariance of a random variable). Let pΩ,F ,Pq be a probabilityspace, let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and let X : Ω Ñ Vbe an F/BpV q-measurable mapping with the property that for all ϕ P V 1 it holds thatE“

|ϕpXq|2‰

ă 8. Then we denote by CovpXq : V 1 ˆ V 1 Ñ K the mapping given byCovpXq “ CovpXpPqq.

Lemma 5.2.39 (Properties of the covariance). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let µ : BpV q Ñ r0,8s be a probability measure with theproperty that for all ϕ P V 1 it holds that

ş

V|ϕpvq|2 µpdvq ă 8. Then Covpµq : V 1 ˆ

V 1 Ñ K is

• nonnegative, i.e., @ϕ P V 1 : pCov µqpϕ, ϕq P r0,8q)

• Hermitian, i.e., @ϕ, ψ P V 1 : pCov µqpϕ, ψq “ pCov µqpψ, ϕq,

• sesquilinear, i.e., @φ, ϕ, ψ P V 1, a P K : pCov µqpaφ` ϕ, ψq “ apCov µqpφ, ψq `pCov µqpϕ, ψq,

• continuous, and

• it holds that

8 ą CovpµqLp2qpV 1,Kq “ supϕ,ψPV 1zt0u

„

|pCov µqpϕ, ψq|

ϕV 1 ψV 1

ď

ż

V

v2V µpdvq. (5.72)


Proof of Lemma 5.2.39. First of all, observe that the nonnegativity, the Hermitian-ity, and the sesquilinearity of Covpµq follow immediately from Definition 5.2.37. Thecontinuity of Covpµq and the fact that

8 ą CovpµqLp2qpV 1,Rq “ supϕ,ψPV 1zt0u

„

|pCov µqpϕ, ψq|

ϕV 1 ψV 1

(5.73)

follow immediately from Corollary 5.2.36. In the next step we observe that Holder’sinequality implies that for all ϕ, ψ P V 1 it holds that

|pCov µqpϕ, ψq| ď

ż

V

ˇ

ˇ

ˇ

ˇ

ϕpvq ´

ż

V

ϕpuqµpduq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ψpvq ´

ż

V

ψpuqµpduq

ˇ

ˇ

ˇ

ˇ

µpdvq

ď

«

ż

V

ˇ

ˇ

ˇ

ˇ

ϕpvq ´

ż

V

ϕpuqµpduq

ˇ

ˇ

ˇ

ˇ

2

µpdvq

ff12 «ż

V

ˇ

ˇ

ˇ

ˇ

ψpvq ´

ż

V

ψpuqµpduq

ˇ

ˇ

ˇ

ˇ

2

µpdvq

ff12

ď

„ż

V

|ϕpvq|2 µpdvq

12 „ż

V

|ψpvq|2 µpdvq

12

ď ϕV 1 ψV 1

ż

V

v2V µpdvq.

(5.74)


5.2.4 Gaussian measures on normed vector spaces

Definition 5.2.40 (One-dimensional Gaussian measures). Let µ : BpRq Ñ r0,8s bea measure with the property that there exist a, b P R such that for all B P BpRq itholds that

µpBq “

ż

txPR : ax`bPBu

1?

2πexp

ˆ

´y2

2

˙

dy. (5.75)

Then we call µ a one-dimensional Gaussian measure.

Definition 5.2.41 (Gaussian measures on possibly infinite dimensional spaces). LetpV, ¨V q be a normed R-vector space and let µ : BpV q Ñ r0,8s be a probability mea-sure with the property that for all ϕ P V 1 it holds that ϕpµqBpRq is a one-dimensionalGaussian measure. Then µ is called Gaussian (on pV, ¨V q).


Example 5.2.42. Let T P p0,8q, m P N, let pΩ,F ,Pq be a probability space, letW : r0, T s ˆ Ω Ñ Rm be a standard Brownian motion with continous sample paths,and let W : Ω Ñ Cpr0, T s,Rmq be the mapping with the property that for all ω P Ω,t P r0, T s it holds that

`

W pωq˘

ptq “ Wtpωq. (5.76)

Then W pPq is a Gaussian measure on`

Cpr0, T s,Rmq, ¨Cpr0,T s,Rmq˘

. To see this

let ϕ P Cpr0, T s,Rmq1 be arbitrary and let PN : Cpr0, T s,Rmq Ñ Cpr0, T s,Rmq, N P

N, pN : Cpr0, T s,Rmq Ñ pRmqN`1, N P N, and ιN : pRmqN`1 Ñ Cpr0, T s,Rmq,N P N, be the mappings with the property that for all v P Cpr0, T s,Rmq, N P N,

pa0, a1, . . . , aNq P pRmqN`1, n P t0, 1, . . . , N ´ 1u, t P rnT

N, pn`1qT

Ns it holds that

`

PNpvq˘

ptq ““

n` 1´ tNT

‰

vpnTNq `

“

tNT´ n

‰

vp pn`1qTN

q, (5.77)

`

ιNpa0, a1, . . . , aNq˘

ptq ““

n` 1´ tNT

‰

an `“

tNT´ n

‰

an`1, (5.78)

pNpvq “`

vp0q, vp TNq, vp2T

Nq, . . . , vpT q

˘

. (5.79)

Observe that for all N P N it holds that PN , pN , and ιN are continuous and thatPN “ ιN ˝ pN . Next let ϕN : pRmqN`1 Ñ R, N P N, be the mappings with theproperty that for all N P N it holds that ϕN “ ϕ ˝ ιN . Moreover, note that for everyN P N it holds that

`

ϕ ˝ PN ˝ W˘

pPq “`

ϕ ˝ PN˘`

W pPq˘

“`

ϕN ˝ pN ˝ W˘

pPq (5.80)

is a Gaussian measure. This shows that for all y P R, N P N it holds that

E

”

exp´

i ¨ y ¨ ϕNppNpW qq¯ı

“ exp

ˆ

i ¨ y Ë”

ϕNppNpW qqı

ý2

2¨ Var

´

ϕNppNpW qq¯

˙

“ exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

. (5.81)

Observe that for all N P N, ω P Ω it holds that limNÑ8 ϕN`

pNpW pωqq˘

“ ϕpW pωqq.This, Lebesgue’s theorem of dominated convergence, and (5.81) imply that for ally P R it holds that

E

”

exp´

i ¨ y ¨ ϕpW q¯ı

“ limNÑ8

E

”

exp´

i ¨ y ¨ ϕNppNpW qq¯ı

“ limNÑ8

exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

“ exp

ˆ

ý2

2¨ limNÑ8

E

”

ˇ

ˇϕpPNpW qqˇ

ˇ

2ı

˙

“ exp

ˆ

ý2

2Ë

”

ˇ

ˇϕpW qˇ

ˇ

2ı

˙

. (5.82)

This proves that ϕ`

W pP q˘

is a one-dimensional Gaussian measure and this shows

that W pPq is indeed a Gaussian measure.


Lemma 5.2.43. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, and letµ : BpHq Ñ r0,8s be a measure. Then µ is Gaussian if and only if for all v P H itholds that Rep〈v, µ〉Hq is a Gaussian measure.


5.2.4.1 Fourier transform of a Gaussian measure

Proposition 5.2.44 (Fourier transform of a Gaussian measure). Let pV, ¨V q be anormed R-vector space and let µ : BpV q Ñ r0,8s be a finite measure. Then µ isGaussian if and only if for all ϕ P V 1 it holds that

ş

V|ϕpvq|2 µpdvq ă 8 and

pFV µqpϕq “ exp

ˆ

i ∫Vϕpvqµpdvq ´ 1

2pCov µqpϕ, ϕq

˙

. (5.83)

Proof of Proposition 5.2.44. First of all, observe that if µ is a Gaussian measure,then it holds for all ϕ P V 1, ξ P R that

ş


pFV µqpξ ¨ ϕq “

ż

V

ei¨ϕpvq¨ξ µpdvq “

ż

R

ei¨x¨ξ`

ϕpµq˘

pdxq

“ exp

ˆ

i ξ ∫Vϕpvqµpdvq ´ ξ2

2pCov µqpϕ, ϕq

˙

.

(5.84)

This proves the “ñ” direction in the statement of Proposition 5.2.44. It thus remainsto prove the “ð” direction in the statement of Proposition 5.2.44. To this end weassume in the following that for all ϕ P V 1 it holds that

ş


pFV µqpϕq “ exp

ˆ

i ∫Vϕpvqµpdvq ´ 1

2pCov µqpϕ, ϕq

˙

. (5.85)

This implies that for all ϕ P V 1, ξ P R it holds that

pFV µqpξ ¨ ϕq “ exp

ˆ

i ξ ∫Vϕpvqµpdvq ´ ξ2

2pCov µqpϕ, ϕq

˙

. (5.86)

This, in turn, proves that for all ϕ P V 1 it holds that ϕpµq is a Gaussian measure onpR,BpRqq. The proof of Proposition 5.2.44 is thus completed.

Corollary 5.2.45 (Covariance of Gaussian measures). Let pV, ¨V q be a separablenormed R-vector space and let µk : BpV q Ñ r0,8s, k P t1, 2u, be Gaussian measureswith the property that Covpµ1q “ Covpµ2q and with the property that for all ϕ P V 1

it holds thatş

Vϕpvqµ1pdvq “

ş

Vϕpvqµ2pdvq. Then µ1 “ µ2.

Corollary 5.2.45 is an immediate consequence from Proposition 5.2.44 and fromLemma 5.2.9.


5.3 Probability measures on Hilbert spaces

5.3.1 Nuclear operators on Hilbert spaces

Below we study Hilbert space valued random variables. The covariance operatorassociated to such a random variable is a nuclear operator. To study such covarianceoperators, we need a few more properties of nuclear operators.

Definition 5.3.1. Let K P tR,Cu and let V be a normed K-vector space. A mapping〈¨, ¨〉 : V ˆV Ñ K is called a scalar product/inner product on V if the following threeproperties are fulfilled:

(i) for all x P V zt0u it holds that 〈x, x〉 P p0,8q,

(ii) for all x, y P V it holds that 〈x, y〉 “ 〈y, x〉, and

(iii) for all λ P K, x, y, z P V it holds that 〈x, y ` λz〉 “ 〈x, y〉` λ 〈x, z〉.

Lemma 5.3.2 (Completeness of the space of nuclear operators). Let K P tR,Cu andlet pV, ¨V q and pW, ¨W q be K-Banach spaces. Then the pair pL1pV,W q, ¨L1pV,W q

q

is a K-Banach space.

Lemma 5.3.2 is, for example, proved as Theorem VI.5.3 (c) in Werner [29]. See,e.g., Lemma VI.5.6 in Werner [29] for the next lemma.

Lemma 5.3.3 (Operators with finite trace). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, let A P L1pHq, let B Ď H be an orthonormal basis of H, and letpvnqnPN Ď H, pwnqnPN Ď H satisfy that for all x P H it holds that

ř8

n“1 vnH wnH ă8 and

Ax “8ÿ

n“1

pwn b vnqpxq. (5.87)

Thenÿ

bPB

|〈b, Ab〉H | ă 8 and8ÿ

n“1

〈vn, wn〉H “ÿ

bPB

〈b, Ab〉H . (5.88)

5.3. PROBABILITY MEASURES ON HILBERT SPACES 133

Proof of Lemma 5.3.3. Observe that the Holder inequality proves that

ÿ

bPB

|〈b, Ab〉H | “ÿ

bPB

ˇ

ˇ

ˇ

ˇ

ˇ

⟨b,

8ÿ

n“1

pwn b vnqpbq

⟩H

ˇ

ˇ

ˇ

ˇ

ˇ

ďÿ

bPB

8ÿ

n“1

|〈b, pwn b vnqpbq〉H |

“ÿ

bPB

8ÿ

n“1

|〈b, wn〉H 〈vn, b〉H | ď8ÿ

n“1

«

ÿ

bPB

|〈b, wn〉H | |〈b, vn〉H |

ff

ď

8ÿ

n“1

«

ÿ

bPB

|〈b, wn〉H |2

ff12 «

ÿ

bPB

|〈b, vn〉H |2

ff12

“

8ÿ

n“1

vnH wnH ă 8.

(5.89)

Moreover, note that

8ÿ

n“1

〈vn, wn〉H “8ÿ

n“1

ÿ

bPB

〈b, vn〉H 〈b, wn〉H “8ÿ

n“1

ÿ

bPB

〈b, wn〉H 〈vn, b〉H

“

8ÿ

n“1

ÿ

bPB

〈b, pwn b vnqpbq〉H “ÿ

bPB

⟨b,

8ÿ

n“1

pwn b vnqpbq

⟩H

“ÿ

bPB

〈b, Ab〉H .(5.90)


Lemma 5.3.3 allow us to introduce the concept of the trace of a nuclear operatoron a Hilbert space.

Definition 5.3.4 (Trace of a nuclear operator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A P L1pHq. Then we denote by traceHpAq P K theelement of K with the property that for all orthonormal basis B Ď H it holds that

traceHpAq “ÿ

bPB

〈b, Ab〉H P K. (5.91)

Lemma 5.3.5. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be K-Hilbert spaces, and let A P LpH,Uq. Then it holds that A˚A P LpHq is nonnegativeand symmetric.

Proof of Lemma 5.3.5. Note that for all v, w P H it holds that

〈v, A˚Aw〉H “⟨rA˚s˚ v,Aw

⟩U“ 〈Av,Aw〉U “ 〈A

˚Av,w〉H . (5.92)

This proves that A˚A is symmetric and that for all v P H it holds that

〈v, A˚Av〉H “ 〈Av,Av〉U “ Av2U ě 0. (5.93)

Hence, we obtain that A˚A is nonnegative. The proof of Lemma 5.3.5 is thus com-pleted.


Lemma 5.3.5 and Definiton 3.4.13 allows us to introduce the absolute value op-erator of a bounded linear operator.

Definition 5.3.6 (The absolute value operator of a bounded linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be K-Hilbert spaces, and letA P LpH,Uq. Then we denote by |A| P LpHq the linear operator given by

|A| “ rA˚As12P LpHq. (5.94)

Lemma 5.3.7 (The trace of the absolute value operator). Let K P tR,Cu, letpV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, let A P LpV,W q, and letB Ď V be an orthonormal basis of V . Then

ř

bPB 〈b, |A| b〉V ă 8 if and only ifA P L1pV,W q and in that case it holds that

ÿ

bPB

〈b, |A| b〉V “ traceV p|A|q “ AL1pV,W q. (5.95)

Lemma 5.3.7 can, e.g., be established by using the theory of singular values; see,e.g., Werner [29].

Proposition 5.3.8 (Properties of the absolute value operator of a bounded linearoperator). Let K P tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbertspaces, and let A P LpV,W q. Then

(i) |A| P LpV q is nonnegative and symmetric,

(ii)ˇ

ˇ |A|ˇ

ˇ “ |A|,

(iii) for all v P V it holds that

AvW “ |A|vV , (5.96)

(iv) for all i P t1, 2u it holds that A P LipV,W q if and only if |A| P LipV q,

(v) A P L2pV,W q if and only if |A|2 “ A˚A P L1pV q and it that case it holds that

A2L2pV,W q“›

›|A|2›

›

L1pV q“ traceV p|A|

2q, (5.97)


(vi) A P L1pV,W q if and only if |A|12P L2pV q and it that case it holds that

AL1pV,W q“›

›|A|12›

›

2

L2pV q“ traceV p|A|q, (5.98)

and

(vii) if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A is symmetric and nonnegative,then |A| “ A.

Proof of Proposition 5.3.8. Definition 3.4.13 ensures that |A| is nonnegative and

symmetric. This shows that ||A|| ““

|A|˚ |A|‰12

““

|A|2‰12

“ |A|. Furthermore,observe that for all v P V it holds that

Av2W “ 〈Av,Av〉W “ 〈v, A˚Av〉V “@

v, |A| |A| vD

V

“@

|A| v, |A| vD

V“ |A|v2V .

(5.99)

This proves (5.96). The identity (5.96), in turn, shows that A P L2pV,W q if and onlyif |A| P L2pV q. Lemma 5.3.7 implies that A P L1pV,W q if and only if there exists anorthonormal basis B Ď V of V such that

ÿ

bPB

〈b, |A| b〉V ă 8. (5.100)

Furthermore, Lemma 5.3.7 proves that |A| P L1pV q if and only if there exists anorthonormal basis B Ď V of V such that

ÿ

bPB

〈b, |A| b〉V ă 8. (5.101)

Combining (5.100) and (5.101) proves that A P L1pV,W q if and only if |A| P L1pV q.Next let B Ď V be an orthonormal basis of V and observe that

ÿ

bPB

Ab2W “ÿ

bPB

〈Ab,Ab〉V “ÿ

bPB

〈b, A˚Ab〉V “ÿ

bPB

⟨b, |A|2 b

⟩V. (5.102)

This and Lemma 5.3.7 prove Item (v). Item (iv), Item (v) and Lemma 5.3.7 proveItem (vi). Moreover, note that if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A issymmetric and nonnegative, then

rA˚As12““

A2‰12“ A. (5.103)



Definition 5.3.9 (Rank-1 operators in Hilbert spaces; cf. Definition 3.4.19). LetK P tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, and letv P V , w P W . Then we denote by w b v P LpV,W q the linear operator with theproperty that for all u P V it holds that

pw b vqpuq “`

w b 〈v, ¨〉V˘

puq “ w 〈v, u〉V . (5.104)

Note, in the setting of Definition 5.3.9, that

w b v P L1pV,W q Ď L2pV,W q “ HSpV,W q Ď LpV,W q (5.105)

and

w b vL1pV,W q“ w b vL2pV,W q

“ w b vLpV,W q “ wW vV . (5.106)

Theorem 5.3.10 (Spectral decomposition for compact operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P KpH,Hq be a symmetric com-pact operator. Then A is a diagonal linear operator in the sense of Definition 3.5.1.

Theorem 5.3.10 is, e.g., proved as Theorem VI.3.2 in Werner [29].

5.3.2 Expectation and covariance operator

Definition 5.3.11 (Covariance operator of a probability measure on a Hilbertspace). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be aprobability measure with the property that for all w P H it holds that

ş

H|〈w, v〉H |

2 µpdvq ă8. Then we denote by CovOppµq P LpHq the unique bounded linear operator suchthat for all v, w P H it holds that

〈v,CovOppµqw〉H “ pCov µq`

〈v, ¨〉H , 〈w, ¨〉H˘

. (5.107)

Lemma 5.3.12 (Properites of covariance operators). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be a probability measure with

ş

Hv2H µpdvq ă

8. Then CovOppµq is a symmetric and nonnegative nuclear operator and it holdsthat

traceHpCovOppµqq “ CovOppµqL1pHqď

ż

H

v2H µpdvq P r0,8q. (5.108)


Proof of Lemma 5.3.12. Nonnegativity and symmetry of CovOppµq follows immedi-ately from Lemma 5.2.39. Next observe that for all orthonormal bases B Ď H of Hit holds thatÿ

bPB

〈b, |CovOppµq| b〉H “ÿ

bPB

〈b,CovOppµqb〉H

“ÿ

bPB

«

ż

H

|〈b, v〉H |2 µpdvq ´

ˇ

ˇ

ˇ

ˇ

ż

H

〈b, v〉H µpdvqˇ

ˇ

ˇ

ˇ

2ff

ďÿ

bPB

ż

H

|〈b, v〉H |2 µpdvq “ sup

BĎBfinite

ÿ

bPB

ż

H

|〈b, v〉H |2 µpdvq

“ supBĎBfinite

ż

H

ÿ

bPB

|〈b, v〉H |2 µpdvq ď sup

BĎBfinite

ż

H

v2H µpdvq

“

ż

H

v2H µpdvq.

(5.109)

Lemma 5.3.7 hence completes the proof of Lemma 5.3.12.

Definition 5.3.13 (Covariance operator of a Hilbert space valued random variable).Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, and letX : Ω Ñ H be an F/BpHq-measurable mapping with the property that for all v P Hit holds that E

“

|〈v,X〉H |2‰

ă 8. Then we denote by CovOppXq P LpHq the linearoperator given by CovOppXq “ CovOp

`

XpPq˘

.

Note, in the setting of Definition 5.3.11, that for all v, w P H it holds that

〈v,CovOppXqw〉H “ E”

`

〈v,X〉H´Er〈v,X〉Hs˘`

〈w,X〉H´Er〈w,X〉Hs˘

ı

. (5.110)

Definition 5.3.14. Let pΩ,F ,Pq be a probability space, let pV, ¨V q be a Banachspace, and let X P L1pP; ¨V q. Then we denote by ErXs P V the element from Vgiven by

ErXs “

ż

Ω

XpωqPpdωq. (5.111)

Note that the integral appearing on the right hand side of (5.111) is a Bochnerintegral; see Section 3.6 for details.


Proposition 5.3.15 (Properties of the covariance operator of a Hilbert space valuedrandom variable). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, and let X P L2pP; ¨Hq. Then

(i) it holds that CovOppXq is a symmetric and nonnegative nuclear operator,

(ii) it holds that

CovOppXq “ E“`

X ÉrXs˘

b`

X ÉrXs˘‰

P L1pHq, (5.112)

and

(iii) it holds that

traceHpCovOppXqq “ E“

X ÉrXs2H‰

“ CovOppXqL1pHqP r0,8q.

(5.113)

5.3.3 Karhunen-Loeve expansion

Theorem 5.3.16 (Karhunen-Loeve expansion). Let pΩ,F ,Pq be a probability space,let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let X P L2pP; ¨Hq, let B Ď Hbe an orthonormal basis of H, and let λ : BÑ r0,8q be a globally bounded functionsuch that for all v P H it holds that CovOppXq v “

ř

bPB λb 〈b, v〉H b. Then

(i) the random variables p〈b,X ÉrXs〉HqbPB are centered and pairwise uncorre-lated,

(ii) it holds for all b P B that Varp〈b,X ÉrXs〉Hq “ Varp〈b,X〉Hq “ λb,

(iii) it holds that

X “ ErXs `ÿ

bPB

〈b,X ÉrXs〉H b, (5.114)

and

(iv) it holds for all B Ď B that›

›

›

›

›

X ´

«

ErXs `ÿ

bPB

〈b,X ÉrXs〉H b

ff›

›

›

›

›

L2pP;¨Hq

“

d

ÿ

bPBzB

λb ă 8. (5.115)


Proof of Theorem 5.3.16. Equation (5.114) follows immediately from the fact thatB is an orthonormal basis of H. Furthermore, note that the random variablesp〈b,X ÉrXs〉HqbPB are centered. Next observe that for all b1, b2 P B it holds that

Er〈b1, X ÉrXs〉H 〈b2, X ÉrXs〉Hs “ 〈b1,CovOppXqb2〉H“

ÿ

bPB

〈b1, λb 〈b, b2〉H b〉H

“ λb1 〈b1, b2〉H .

(5.116)

This implies that the random variables p〈b,X ÉrXs〉HqbPB are pairwise uncorre-lated and it shows that for all b P B it holds that

Varp〈b,X〉Hq “ Varp〈b,X ÉrXs〉Hq “ λb. (5.117)

Combining this with (5.114) proves that for all B Ď B it holds that

›

›

›

›

›

X ´

«

ErXs `ÿ

bPB

〈b,X ÉrXs〉H b

ff›

›

›

›

›

L2pP;¨Hq

“

›

›

›

›

›

›

ÿ

bPBzB

〈b,X ÉrXs〉H b

›

›

›

›

›

›

L2pP;¨Hq

“

d

ÿ

bPBzB

E“

|〈b,X ÉrXs〉H |2‰

“

d

ÿ

bPBzB

λb.

(5.118)


5.3.4 Gaussian measures on Hilbert spaces

5.3.4.1 Karhunen-Loeve expansion

Definition 5.3.17 (Gaussian distributed random variables). Let pΩ,F ,Pq be a prob-ability space, let pV, ¨V q be a normed R-vector space, and let X : Ω Ñ V be anF/BpV q-measurable mapping with the property that XpPq is a Gaussian measure.Then X is called Gaussian distributed.


Corollary 5.3.18 (Karhunen-Loeve expansion for Gaussian distributed randomvariables). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be a separableR-Hilbert space, let X P L2pP; ¨Hq be Gaussian distributed, let B Ď H be an or-thonormal basis of H, and let λ : BÑ r0,8q be a globally bounded function such thatfor all v P H it holds that CovOppXq v “

ř

bPB λb 〈b, v〉H b. Then the random vari-ables 1?

λb〈b,X ÉrXs〉H , b P λ´1pp0,8qq, are independent identically distributed

(i.i.d.) standard normal random variables and it holds P-a.s. that

X “ ErXs `ÿ

bPλ´1pp0,8qq

a

λb

„

〈b,X ÉrXs〉H?λb

b. (5.119)

Proof of Corollary 5.3.18. First of all, observe that Theorem 5.3.16 together withthe assumption that XpPq is a Gaussian measure imply that the random variables

1?λb〈b,X ÉrXs〉H , b P λ´1pp0,8qq, are i.i.d. standard normal random variables.

Moreover, Theorem 5.3.16 ensures that for all b P λ´1pt0uq it holds that

E“

|〈b,X ÉrXs〉H |2‰

“ Varp〈b,X ÉrXs〉Hq “ λb “ 0. (5.120)

This proves that for all b P λ´1pt0uq it holds P-a.s. that

〈b,X ÉrXs〉H “ 0. (5.121)

This shows that it holds P-a.s. that

X ÉrXs “ÿ

bPλ´1pp0,8qq

〈b,X ÉrXs〉H b. (5.122)

Equation (5.122) implies (5.119). The proof of Corollary 5.3.18 is thus completed.

5.3.4.2 Construction of Gaussian measures on Hilbert spaces

In Theorem 5.3.21 below we establish the existence of Gaussian measures on Hilbertspaces. In the proof of Theorem 5.3.21 we use the fact that the set of Cauchysequences of a sequence of strongly measurable mappings is a measurable set; seeLemma 5.3.19 below.

Lemma 5.3.19 (Cauchy sequence). Let pΩ,Fq be a measurable space, let pE, dEqbe a metric space, and let Xn : Ω Ñ E, n P N, be F/pE, dEq-strongly measurablemappings. Then it holds that

ω P Ω: pXnpωqqnPN is a Cauchy-sequence(

P F . (5.123)


Proof of Lemma 5.3.19. First of all, note that the assumption that Xn, n P N,are F/pE, dEq-strongly measurable ensures that for all n,m P N it holds thatpXn, Xmq : Ω Ñ E ˆ E is F/BpE ˆ Eq-measurable. The continuity of the map-ping dE : E ˆ E Ñ r0,8q hence implies that for all n,m P N it holds that thefunction

Ω Q ω ÞÑ dEpXnpωq, Xmpωqq P r0,8q (5.124)

is F/Bpr0,8qq-measurable. This ensures that for all k, n,m P N it holds that

tdEpXn, Xmq ă 1ku P F . (5.125)

This implies that

ω P Ω: pXnpωqqnPN is a Cauchy-sequence(

“

ω P Ω: @ ε P p0,8q : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă ε(

“

ω P Ω: @ k P N : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă 1k(

“ XkPN YNPN Xn,mPNXrN,8q tω P Ω: dEpXnpωq, Xmpωqq ă 1ku

“ XkPN YNPN Xn,mPtN,N`1,... u tdEpXn, Xmq ă 1kulooooooooooomooooooooooon

PF

P F .

(5.126)

This completes the proof of Lemma 5.3.19.

The next result, Corollary 5.3.20, is a direct consequence of Proposition 5.2.44above.

Corollary 5.3.20 (Fourier transform of a Gaussian measure on a Hilbert space). LetpH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be a finite measure.Then µ is Gaussian if and only if for all v P H it holds that

ş

H|〈v, w〉H |

2 µpdwq ă 8and

pFV µqp〈v, ¨〉Hq “ exp`

i ∫H 〈v, w〉H µpdwq ´12〈v,CovOppµq v〉H

˘

. (5.127)

The next result, Theorem 5.3.21, establishes the existence of a Gaussian measureon a Hilbert space with a given mean vector and a given nuclear covariance operator.

Theorem 5.3.21 (Gaussian measures on Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq bean R-Hilbert space, let v P H, and let Q P L1pHq be a nonnegative and symmetricnuclear operator. Then there exists a Gaussian measure Nv,Q : BpHq Ñ r0,8s withthe property that CovOppNv,Qq “ Q and with the property that for all w P H it holdsthat 〈w, v〉H “

ş

H〈w, x〉H Nv,Qpdxq.


Proof of Theorem 5.3.21. Theorem 5.3.10 proves that there exists an orthonormalbasis B Ď H of H and a globally bounded function λ : B Ñ r0,8q such that for allx P H it holds that

Qx “ÿ

bPB

λb 〈b, x〉H b. (5.128)

Proposition 3.5.5 proves thatř

bPB |λb| ă 8. This and Lemma 3.2.16 show thatthe set λ´1pp0,8qq is at most countable. W.l.o.g. we assume that λ´1pp0,8qq iscountable. Hence, there exist a sequence pbnqnPN Ď λ´1pp0,8q with the propertythat for all n,m P N with n ‰ m it holds that bn ‰ bm and with the property that

tbn P B : n P Nu “ λ´1pp0,8qq. (5.129)

Next let pΩ,F ,Pq be a probability space and let Yn : Ω Ñ R, n P N, be i.i.d. standardnormal random variables. Observe that such a probability space does indeed exist.In the next step let XN : Ω Ñ H, N P N, be the mappings with the property thatfor all N P N it holds that

XN “

Nÿ

n“1

a

λbnYnbn. (5.130)

Note that for all N P N it holds that XN is F/pH, ¨Hq-strongly measurable.Lemma 5.3.19 and the completeness of pH, ¨Hq hence prove that the set

A “ tω P Ω: pXNpωqqNPN Ď H is convergent u (5.131)

is in F . This shows that for all N P N it holds that

Ω Q ω ÞÑ 1Apωq ¨XNpωq P H (5.132)

is F/pH, ¨Hq-strongly measurable. Moreover, observe that for all p P r1,8q it holdsthat

›

›

›

›

supNPN

XN2H

›

›

›

›

LppP;|¨|q

“

›

›

›

›

›

supNPN

«

Nÿ

n“1

λbn |Yn|2

ff›

›

›

›

›

LppP;|¨|q

“

›

›

›

›

›

limNÑ8

«

Nÿ

n“1

λbn |Yn|2

ff›

›

›

›

›

LppP;|¨|q

“ limNÑ8

›

›

›

›

›

Nÿ

n“1

λbn |Yn|2

›

›

›

›

›

LppP;|¨|q

ď limNÑ8

Nÿ

n“1

λbn›

›|Yn|2›

›

LppP;|¨|q“›

›|Y1|2›

›

LppP;|¨|q

8ÿ

n“1

λbn ă 8.

(5.133)


In the next step let X : Ω Ñ H be the mapping with the property that for all ω P Ωit holds that

Xpωq “ v ` 1Apωq”

limNÑ8

XNpωqı

“ v ` limNÑ8

r1Apωq ¨XNpωqs . (5.134)

Combining (5.132) and Theorem 2.4.7 proves that X is F/pH, ¨Hq-strongly mea-surable. Furthermore, note that for all N P N it holds that

E

«

supM1,M2PtN,N`1,... u

XM1 ´XM22H

ff

“ E

«

8ÿ

n“N`1

›

›

›

a

λbnYnbn

›

›

›

2

H

ff

“

8ÿ

n“N`1

E

„

›

›

›

a

λbnYnbn

›

›

›

2

H

“

8ÿ

n“N`1

λbn E“

|Yn|2H

‰

“

8ÿ

n“N`1

λbn ă 8.

(5.135)

This implies that PpAq “ 1. Moreover, (5.133) ensures that for all p P p0,8q it holdsthat

X ´ vHLppP;|¨|q “

›

›

›

›

›

›

›limNÑ8

p1AXNq

›

›

›

H

›

›

›

›

LppP;|¨|q

“

›

›

›limNÑ8

p1A XNHq

›

›

›

LppP;|¨|qď

›

›

›

›

supNPN

XNH

›

›

›

›

LppP;|¨|q

ă 8.

(5.136)

This, in particular, implies that for all p P p0,8q it holds that

E

„

XpH ` supNPN

XNpH

ă 8. (5.137)

Uniform integrability and the fact that PpAq “ 1 hence prove that

E“

X‰

“ E

”

v ` limNÑ8

p1AXNq

ı

“ v `E”

limNÑ8

p1AXNq

ı

“ v ` limNÑ8

Er1AXN s “ v ` limNÑ8

ErXN s

“ v ` limNÑ8

«

Nÿ

n“1

a

λbn ErYns bn

ff

“ v.

(5.138)

In addition, uniform integrability, the fact that PpAq “ 1, and (5.137) ensure that


for all x, y P H it holds that

〈x,CovOppXqy〉H “ 〈x,CovOppX ´ vqy〉H “ Er〈x,X ´ v〉H 〈y,X ´ v〉Hs“ lim

NÑ8Er〈x,1AXN〉H 〈y,1AXN〉Hs “ lim

NÑ8Er〈x,XN〉H 〈y,XN〉Hs

“ limNÑ8

〈x,CovOppXNqy〉H “ limNÑ8

«

Nÿ

n“1

λbn 〈x, bn〉H 〈y, bn〉H E“

|Yn|2‰

ff

“ limNÑ8

«

Nÿ

n“1

λbn 〈x, bn〉H 〈y, bn〉H

ff

“ limNÑ8

⟨x,

Nÿ

n“1

λbn 〈bn, y〉H bn

⟩H

“ 〈x,Qy〉H .

(5.139)

This implies thatCovOppXq “ Q. (5.140)

Next note that Lebesgue’s theorem of dominated convergence and the fact thatPpAq “ 1 imply that for all w P H it holds that

`

FHXpPq˘

p〈w, ¨〉Hq “ E“

ei〈w,X〉H‰

“ ei〈w,v〉H E“

ei〈w,X´v〉H‰

“ ei〈w,v〉H limNÑ8

E“

ei〈w,1AXN 〉H‰


E“

ei〈w,XN 〉H‰


e´12〈w,CovOppXN qw〉H “ ei〈w,v〉H´

12〈w,CovOppXqw〉H .

(5.141)

This, (5.137), and Corollary 5.3.20 imply that XpPq is a Gaussian measure. Combin-ing this with (5.138) and (5.140) establishes the existence of a probability measureXpPq with the desired properties. The proof of Theorem 5.3.21 is thus completed.

5.3.4.3 Karhunen-Loeve expansion for Brownian motion

Exercise 5.3.22 (An ordinary differential equation of second order). Let a P Rzt0u,T P p0,8q and let v : r0, T s Ñ R be a twice continuously differentiable function withthe property that for all t P r0, T s it holds that v2ptq “ a vptq. Prove that for allt P r0, T s it holds that

˜

vptq

v1ptq

¸

“

˜

1 1?a ´

?a

¸˜

et?a 0

0 e´t?a

¸˜

1 1?a ´

?a

¸´1 ˜

vp0q

v1p0q

¸

(5.142)

and prove that there exist A,B P C such that for all t P r0, T s it holds that vptq “Ae

?at `Be´

?at.


Definition 5.3.23. Let d P N, A P BpRdq. Then we denote by BorelA : BpAq Ñr0,8s the Lebesgue-Borel measure on A.

Let us illustrate Definition 5.3.23 through a simple example. Note that for alla, b, α, β P R with a ď α ď β ď b it holds that

Borelra,bsprα, βsq “ β ´ α. (5.143)

Theorem 5.3.24 (Karhunen-Loeve expansion for Brownian motion). Let pΩ,F ,Pqbe a probability space, let T P p0,8q, let W : r0, T s ˆΩ Ñ R be a standard Brownianmotion with continuous sample paths, let W : Ω Ñ L2pBorelp0,T q; |¨|Rq be the mappingwith the property that for all ω P Ω and Borelp0,T q-almost all t P r0, T s it holds that

pW pωqqptq “ Wtpωq, and let ek P L2pBorelp0,T q; |¨|Rq, k P N, be the vectors withthe property that for all k P N and Borelp0,T q-almost all t P p0, T q it holds that

ekptq “?

2?T

sin`

pk ´ 12qπtT

˘

. Then

(i) W is a Gaussian distributed random variable,

(ii) it holds for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q that

`

CovOppW q v˘

ptq “

ż T

0

mintt, su vpsq ds “

ż t

0

ż T

s

vpuq du ds

“

ż T

0

ErWtWss vpsq ds,

(5.144)

(iii) it holds that

σP`

CovOppW q˘

“

"

T 2

r12πs2,

T 2

r32πs2,

T 2

r52πs2, . . .

*

“

"

T 2

π2 pk ´ 12q2 P p0,8q : k P N

*

,

(5.145)

(iv) the set tek : k P Nu is an orthonormal basis of L2pBorelp0,T q; |¨|Rq,

(v) it holds for all v P L2pBorelp0,T q; |¨|Rq that

CovOppW q v “8ÿ

n“1

T 2

π2 rn´ 12s2 〈en, v〉L2pBorelp0,T q;|¨|Rq

en, (5.146)


(vi) the random variables πTrk ´ 12s xek, W yL2pBorelp0,T q;|¨|Rq

, k P N, are i.i.d. standardnormal random variables, and

(vii) it holds that

W “

8ÿ

k“1

Tπ rk´12s

”

π rk´12s

Txek, W yL2pBorelp0,T q;|¨|Rq

ı

ek. (5.147)

Proof of Theorem 5.3.24. First of all, note that for all v P Cpr0, T s,Rq it holds that

ż T

0

vpsqWs ds “ limNÑ8

«

N´1ÿ

n“0

vpnTNqWnT

N

TN

ff

. (5.148)

This proves that for all v P Cpr0, T s,Rq it holds thatşT

0vpsqWs ds is Gaussian

distributed. This and the fact that Cpr0, T s,Rq is dense in L2pBorelp0,T q; |¨|Rq implies

that for all v P L2pBorelp0,T q; |¨|Rq it holds thatşT

0vpsqWs ds is Gaussian distributed.

It thus holds that W is a Gaussian distributed random variable. Next observe thatfor all v, w P L2pBorelp0,T q; |¨|Rq it holds that

xw,CovOppW qvyL2pBorelp0,T q;|¨|Rq“ E

„ż T

0

vpsqWs ds

ż T

0

wpuqWu du

“

ż T

0

wpuq

ż T

0

ErWsWus vpsq ds du “

ż T

0

wpuq

ż T

0

minps, uq vpsq ds du.

(5.149)

This and Fubini’s theorem show that for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q it holds that

`

CovOppW q v˘

ptq “

ż T

0

ErWsWts vpsq ds “

ż T

0

mints, tu vpsq ds

“

ż T

0

ż mintu,tu

0

vpuq ds du “

ż t

0

ż T

0

1tsďuu vpuq du ds

“

ż t

0

ż T

s

vpuq du ds.

(5.150)

This proves (ii). In the next step let µ P R and let v : p0, T q Ñ R be an Bpp0, T qq/BpRq-measurable function with the property that

şT

0|vpsq|2 ds “ 1 and with the property

thatµ ¨ v “ CovOppW q v. (5.151)


Equation (5.151) implies that for Borelr0,T s-almost all t P p0, T q it holds that

µ ¨ vptq “

ż T

0

minps, tq vpsq ds. (5.152)

Next let w : r0, T s Ñ R be the function with the property that for all t P r0, T s itholds that

wptq “

ż t

0

ż T

s

vpuq du ds. (5.153)

Note that w is continuously differentiable and observe that w1 : r0, T s Ñ R is abso-lutely continuous. Moreover, note that (ii) implies that for all t P r0, T s it holds that

wptq “

ż T

0


Combining this with (5.152) proves that for Borelr0,T s-almost all t P r0, T s it holdsthat

µ ¨ vptq “ wptq. (5.155)

Next note that (5.155) implies that if µ “ 0, then it holds for all t P r0, T s that0 “ wptq “ w1ptq “ w2ptq and this shows that for Borelr0,T s-almost all t P r0, T s itholds that

0 “ w2ptq “ ´vptq. (5.156)

Equation (5.156) contradicts to the assumption thatşT

0|vpsq|2 ds “ 1 ą 0 and this

proves that µ ‰ 0. Next let v : p0, T q Ñ R be a continuously differentiable functiondefined by vptq :“ 1

µwptq for all t P r0, T s. Equation (5.155) then shows that for

Borelr0,T s-almost all t P p0, T q it holds that

vptq “ vptq (5.157)

and (5.153) and (5.154) hence imply that for all t P r0, T s it holds that

µ ¨ vptq “ wptq “

ż t

0

ż T

s

vpuq du ds “

ż T

0


This proves that w and v are twice continuously differentiable with the property thatfor all t P r0, T s it holds that

v2ptq “1

µw2ptq “

´1

µvptq. (5.159)


Exercise 5.3.22 hence proves that there exist A,B P C such that for all t P r0, T s itholds that vptq “ A exp

`

ta

´1µ˘

` B exp`

´ ta

´1µ˘

. This together with the fact

that vp0q “ wp0qµ“ 0 shows that for all t P r0, T s it holds that

vptq “ A”

exp´

ta

´1µ

¯

´ exp´

´ta

´1µ

¯ı

. (5.160)

The identity v1pT q “ w1pT qµ“ 0 and the assumption that

şT

0|vpsq|2 ds “ 1 ą 0 hence

prove that

exp´

2Ta

´1µ

¯

“ ´1. (5.161)

This implies that µ ą 0 and that there exists a k P N such that 2Ta

1µ “ 2πk ´ π.Hence, we obtain that

µ “ T 2

π2rk´12s2 . (5.162)

Putting this into (5.160) proves that for all t P r0, T s it holds that

vptq “ 2iA sin`“

k ´ 12

‰

tπT

˘

. (5.163)

This and the assumption thatşT

0|vpsq|2 ds “ 1 implies that

1 “

ż T

0

|vpsq|2 ds “ 4 |A|2ż T

0

ˇ

ˇsin`“

k ´ 12

‰

sπT

˘ˇ

ˇ

2ds

“ 2T |A|2ż 1

0

ˇ

ˇ

?2 sin

`

rk ´ 12s πs

˘ˇ

ˇ

2ds “ 2T |A|2 .

(5.164)

This and (5.163) prove that there exists a z P t´1, 1u such that for all t P r0, T s itholds that

vptq “ z?

2?T

sin`“

k ´ 12

‰

tπT

˘

“ z ekptq. (5.165)

Furthermore, observe that for all k P N, t P r0, T s it holds that

d2

dt2

„ż T

0

mints, tu ekpsq ds

“d2

dt2

„ż t

0

ż T

s

ekpuq du ds

“d

dt

„ż T

t

ekpsq ds

“ ´ekptq

“ T 2

π2rk´12s2 ¨ e

2kptq “

d2

dt2

”

T 2

π2rk´12s2 ¨ ekptq

ı

.

(5.166)


This together with the fact that for all k P N it holds that

T 2

π2rk´12s2 ¨ ekp0q “ 0 “

ż T

0

minps, 0q ekpsq ds and

T 2

π2rk´12s2 ¨ e

1kpT q “

Tπrk´12s

¨?

2?T¨ cos

`“

k ´ 12

‰

TπT

˘

“ T?

2πrk´12s

?T¨ cos

`“

k ´ 12

‰

π˘

“ 0 “

„

d

dt

ż t

0

ż T

s

ekpuq du ds

t“T

“

„

d

dt

ż T

0

minps, tq ekpsq ds

t“T

(5.167)

proves that for all k P N it holds that

T 2

π2rk´12s2 ¨ ek “ CovOppW q ek. (5.168)

Combining this with (5.162) proves (iii). Moreover, (5.168), (5.165) and Theo-rem 5.3.10 imply (iv) and (v). Finally, (iv), (v) and Corollary 5.3.18 prove (vi)–(vii).The proof of Theorem 5.3.24 is thus completed.

1 function [ Preimage , BM] = KLE Brownian Motion (T,N, Grid )2 Preimage = ( 0 :T/Grid :T) ;3 BM = Preimage ∗0 ;4 for n=1:N5 s q r t e i g e n v n = T/(n ´ 1/2)/ pi ;6 e i g e n f n = sqrt (2/T)∗ sin ( Preimage/ s q r t e i g e n v n ) ;7 BM = BM + s q r t e i g e n v n ∗ e i g e n f n ∗ randn ;8 end9 end

Matlab code 5.1: A Matlab function for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 10 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;


9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f

Matlab code 5.2: A Matlab code for the approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 100 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;

10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f

Matlab code 5.3: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 1000 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;

10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;


13 hold o f f

Matlab code 5.4: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2

Figure 5.1: Results of calls of the Matlab codes 5.2–5.4.

Chapter 6

Stochastic processes with values ininfinite dimensional spaces

6.1 Hilbert space valued stochastic processes

6.1.1 Standard Wiener processes

Definition 6.1.1. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let Q PL1pHq be nonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a filtered probabilityspace, let W : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-adapted stochastic process with theproperties

(i) that W0 “ 0,

(ii) that W has continuous sample paths,

(iii) that for all t1, t2 P r0, T s with t1 ď t2 it holds that σΩpWt2 ´Wt1q and Ft1 areindependent, and

(iv) that for all t1, t2 P r0, T s with t1 ď t2 it holds that pWt2 ´Wt1qpPq “ N0,Qpt2´t1q.

Then W is called a standard Q-Wiener process with respect to (w.r.t.) pFtqtPr0,T s.

Definition 6.1.2. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, letQ P L1pHq be nonnegative and symmetric, let pΩ,F ,Pq be a probability space, andlet W : r0, T s ˆ Ω Ñ H be a standard Q-Wiener process w.r.t. pFWt qtPr0,T s. Then Wis called a standard Q-Wiener process.

153

154 CHAPTER 6. STOCHASTIC PROCESSES

Theorem 6.1.3. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, andlet Q P L1pHq be nonnegative and symmetric. Then there exist a probability spacepΩ,F ,Pq and a standard Q-Wiener process W : r0, T s ˆ Ω Ñ H.

Theorem 6.1.3 can, e.g., be proved by using a Karhunen-Loeve expansion similaras in the proof of Theorem 5.3.21.

6.1.2 Pseudo inverse

Lemma 6.1.4. Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert

spaces, and let A P LpH1, H2q. Then the mapping A|KernpAqJ : KernpAqJ Ñ H2 isinjective and it holds that impAq “ im

`

A|KernpAqJ˘

.

Proof of Lemma 6.1.4. First of all, recall that

KernpAqJ “

v P H1 :“

@u P KernpAq : 〈v, u〉H1“ 0

‰(

(6.1)

is a K-vector subspace of H1. The mapping A|KernpAqJ : KernpAqJ Ñ H2 is thus alinear mapping from KernpAqJ to H2. It thus holds that

A|KernpAqJ is injective ô Kern`

A|KernpAqJ˘

“ t0u. (6.2)

Next note that

Kern`

A|KernpAqJ˘

“

v P KernpAqJ : A|KernpAqJpvq “ 0(

“

v P KernpAqJ : Av “ 0(

“

v P KernpAqJ : v P KernpAq(

“ KernpAqJ XKernpAq “ t0u.

(6.3)

Combining this with (6.2) proves that A|KernpAqJ is injective. Moreover, observe that

impAq “ ApH1q “ tAv P H2 : v P H1u

“

A“

PKernpAq rvs ` PKernpAqJ rvs‰

P H2 : v P H1

(

“

APKernpAqJ rvs P H2 : v P H1

(

“

APKernpAqJ rvs P H2 : v P KernpAqJ(

“

Av P H2 : v P KernpAqJ(

“ im`

A|KernpAqJ˘

.

(6.4)


6.1. HILBERT SPACE VALUED STOCHASTIC PROCESSES 155

Lemma 6.1.4 allows us to introduce the following concept.

Definition 6.1.5. Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbertspaces, and let A P LpH1, H2q. Then we denote by A´1 : impAq Ñ H1 the linearoperator with the property that for all v P impAq it holds that

A´1pvq “ A|´1

KernpAqJpvq (6.5)

and we call A´1 the pseudo inverse of A.

The next exercise will help us to get more familiar with the pseudo inverse.

Exercise 6.1.6. Let A : L2pBorelp0,1q; |¨|Rq Ñ R be the linear mapping with the prop-erty that for all v P L2pBorelp0,1q; |¨|Rq it holds that

Av “

ż 1

0

vpxq dx. (6.6)

Specify DpA´1q, impA´1q, rangepA´1q, and A´1v, v P DpA´1q, explicity. Show thatyour specifications are indeed correct.

In the following proposition we present an important property of the pseudoinverse of a bounded linear operator.

Proposition 6.1.7 (Minimality property of the pseudo inverse). Let K P tR,Cu,let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces, let A P LpH1, H2q, and letv P impAq “ DpA´1q. Then A´1vH1

“ infuPA´1ptvuq uH1and

A´1v(

“

#

w P H1 :

«

Aw “ v and wH1“ inf

uPH1,Au“v

uH1

ff+

“

"

w P A´1ptvuq : wH1

“ infuPA´1ptvuq

uH1

*

“ A´1ptvuq X

“

KernpAqJ‰

.

(6.7)

Proof of Proposition 6.1.7. First of all, note that Lemma 6.1.4 and the definition ofthe pseudo inverse prove that

A´1v(

“

!

A|´1KernpAqJ

pvq)

“

w P KernpAqJ : Aw “ v(

““

KernpAqJ‰

X A´1ptvuq.

(6.8)


Next recall that KernpAq Ď H1 is a closed subspace of H1. Definition 3.3.12 thusshows that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that

A|KernpAqJ“

PKernpAqJ rws‰

“ A“

PKernpAqJ rws‰

“ A“

w ´ PKernpAq rws‰

“ Aw “ v.(6.9)

The fact that A|KernpAqJ is injective (see Lemma 6.1.4) hence proves that for allw P A´1ptvuq “ tu P H1 : Au “ vu it holds that

PKernpAqJ rws “ A´1v. (6.10)

This implies that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that

wH1“›

›PKernpAq rws ` PKernpAqJ rws›

›

H1

“

b

›

›PKernpAq rws›

›

2

H1`›

›PKernpAqJ rws›

›

2

H1

“

b

›

›PKernpAq rws›

›

2

H1` A´1pvq2H1

ě›

›A´1pvq

›

›

2

H1.

(6.11)

This and the fact that A´1pvq P A´1ptvuq prove that

infuPA´1ptvuq

uH1“›

›A´1pvq

›

›

H1. (6.12)

This, (6.11), and (6.8) imply that"

w P A´1ptvuq : wH1

“ infuPA´1ptvuq

uH1

*

“

!

w P A´1ptvuq : wH1

“›

›A´1pvq

›

›

H1

)

“

"

w P A´1ptvuq :

b

›

›PKernpAq rws›

›

2

H1` A´1pvq2H1

“›

›A´1pvq

›

›

H1

*

“

!

w P A´1ptvuq :

›

›PKernpAq rws›

›

2

H1“ 0

)

“

w P A´1ptvuq : PKernpAq rws “ 0

(

“ A´1ptvuq X

“

KernpAqJ‰

“

A´1pvq

(

.

(6.13)

This, (6.8), and (6.12) complete the proof of Proposition 6.1.7.

The pseudo inverse allows us to define a Hilbert space structure on the imageof a bounded linear operator on Hilbert spaces. This is the subject of the nextproposition.

Proposition 6.1.8 (Image Hilbert space). Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq,k P t1, 2u, be K-Hilbert spaces, and let A P LpH1, H2q. Then it holds that the triple`

impAq, 〈A´1p¨q, A´1p¨q〉H1, A´1p¨qH1

˘

is an K-Hilbert space.

6.2. STOCHASTIC INTEGRATION 157

Proof of Proposition 6.1.8. First of all, note that the fact that A´1 : impAq Ñ H1 isa linear operator implies that the triple

`

impAq, 〈A´1p¨q, A´1p¨q〉H1, A´1p¨qH1

˘

is aK-inner product space. It thus remains to prove that pimpAq, A´1p¨qH1

q is com-plete. To see this let pvnqnPN Ď impAq be a Cauchy sequence in pimpAq, A´1p¨qH1

q.Then A´1pvnq P KernpAqJ, n P N, is a Cauchy sequence in pKernpAqJ, ¨H1

q. Com-pleteness of pKernpAqJ, ¨H1

q hence proves that there exists a vector w P KernpAqJ

such that limnÑ8 w ´ A´1pvnqH1

“ 0. This proves that Aw P impAq satisfieslimnÑ8 A

´1pAw ´ vnqH1“ 0. The proof of Proposition 6.1.8 is thus completed.

6.2 Stochastic integration with respect to infinite

dimensional Wiener processes

The following presentations are similiar to the presentations in Section 2.3 in Prevot& Rockner [24] and in Chapter 3 in the lecture notes of the course Numerical Analysisof Stochastic Ordinary Differential Equations.

6.2.1 Filtrations

Definition 6.2.1 (Filtration). Let pΩ,Fq be a measurable space, let T Ď`

R Y

t´8,8u˘

be a set, and let pFtqtPT be a family of sigma-algebras on Ω with the propertythat for every t1, t2 P T with t1 ď t2 it holds that

Ft1 Ď Ft2 Ď F . (6.14)

Then pFtqtPT is called a filtration on pΩ,Fq.

Definition 6.2.2 (Filtrations associated to a filtration). Let T Ď`

RYt´8,8u˘

bea set and let pΩ,Fq be a measurable space with a filtration pFtqtPT. Then we denoteby pF´t qtPT and pF`t qtPT the filtrations on pΩ,Fq with the property that for all t P Tit holds that

F´t “

#

σΩ

`

YsPTXp´8,tqFs˘

: t ą infpTqFt : t “ infpTq

(6.15)

and

F`t “

#

XsPTXpt,8qFs : t ă suppTqFt : t “ suppTq

. (6.16)


Observe, in the setting of Definition 6.2.2, that for all t P T it holds that F´t ĎFt Ď F`t .

Definition 6.2.3. Let T Ď`

R Y t´8,8u˘

be a set and let pΩ,Fq be a measurablespace with a filtration pFtqtPT which satisfies that for all t P T it holds that Ft “F´t (Ft “ F`t ). Then we say that the filtration pFtqtPT is left-continuous (right-continuous).

Next we present the notion of a normal filtration (cf., e.g., Definition 2.1.11 in[24]) and of a stochastic basis (cf. Appendix E in [24]).

Definition 6.2.4 (Normal filtration). Let T P r0,8q and let pΩ,F ,Pq be a probabilityspace with a filtration pFtqtPr0,T s which satisfies

(i) that tA P F : P pAq “ 0u Ď F0 and

(ii) that pFtqtPr0,T s is right-continuous.

Then we say that pFtqtPr0,T s is normal (or also that pFtqtPr0,T s fulfills the usual con-ditions).

Definition 6.2.5 (Stochastic basis). Let T P r0,8q and let pΩ,F , P q be a probabilityspace with a normal filtration pFtqtPr0,T s. Then the quadrupel pΩ,F ,P, pFtqtPr0,T sq iscalled a stochastic basis.

Let us also point out that if T P r0,8q is a real number and if pΩ,F , P qis a probability space with a filtration pFtqtPr0,T s, then sometimes the quadrupelpΩ,F ,P, pFtqtPr0,T sq is called a stochastic basis in the literature although pFtqtPr0,T s isnot necessarily normal.

6.2.2 Lenglart’s inequality

Definition 6.2.6 (Random time). Let T Ď pRY t´8,8uq be a set, let pΩ,F ,Pq bea probability space, and let τ : Ω Ñ T be an F/BpTq-measurable mapping. Then τ iscalled a random time.

Observe, in the setting of Definition 6.2.6, that for every t P T it holds thattτ ď tu P F .

Definition 6.2.7 (Stopping time). Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pqbe a probability space with a filtration pFtqtPT, and let τ : Ω Ñ T be a mapping withthe property that for all t P T it holds that tτ ď tu P Ft. Then τ is called an pFtqtPT-stopping time.


A stopping time on a filtered probability space also induces a sigma-algebra. Thisis the subject of the next definition.

Definition 6.2.8. Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pq be a probabilityspace with a filtration pFtqtPT, and let τ : Ω Ñ T be an pFtqtPT-stopping time. Thenwe denote by Fτ the set given by

Fτ “ tA P pYtPTFtq : p@ t P T : AX tτ ď tu P Ftqu (6.17)

and we call Fτ the sigma-algebra at the stopping time τ .

Exercise 6.2.9. Let T Ď pRYt´8,8uq be a set, let pΩ,F ,Pq be a probability spacewith a filtration pFtqtPT, and let τ, ρ : Ω Ñ T be pFtqtPT-stopping times. Prove thenthat mintτ, ρu is an pFtqtPT-stopping time.

In (6.20) in the following result, Proposition 6.2.10, we prove a powerful inequalitywhich is known as Lenglart inequality in the literature. Proposition 6.2.10 and itsproof are extensions of Problem 1.4.15, Remark 1.4.17 and Solution 4.15 in Section1.6 in [16].

Proposition 6.2.10 (Lenglart inequality). Let pΩ,F ,Pq be a probability space witha filtration pFtqtPr0,8q, let X, Y : r0,8qˆΩ Ñ r0,8q be pFtqtPr0,8q-adapted stochasticprocesses with continuous sample paths such that for all bounded pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that E

“

Xτ

‰

ď E“

suptPr0,τ s Yt‰

. Then for all ε, δ P p0,8qand all pFtqtPr0,8q-stopping times τ : Ω Ñ r0,8q it holds that

P`

suptPr0,τ sXt ě ε˘

ď 1εE“

suptPr0,τ s Yt‰

, (6.18)

P`

suptPr0,τ sXt ě ε, suptPr0,τ s Yt ă δ˘

ď 1εE“

min

δ, suptPr0,τ s Yt(‰

, (6.19)

P`


ď 1εE“

min


` P`

suptPr0,τ s Yt ě δ˘

, (6.20)

E“

min

ε, suptPr0,τ sXt

(‰

ď

”

2?ε` ε?

δ

ı

ˇ

ˇE“

min

δ, suptPr0,τ s Yt(‰ˇ

ˇ

12, (6.21)

E“

min

1, suptPr0,τ sXt

(‰

ď 3ˇ

ˇE“

min

1, suptPr0,τ s Yt(‰ˇ

ˇ

12. (6.22)

Proof of Proposition 6.2.10. Throughout this proof let ρXε : Ω Ñ r0,8s, ε P r0,8q,and ρYε : Ω Ñ r0,8s, ε P r0,8q, be the mappings with the property that for allε P r0,8q it holds that

ρXε “ inf`

t P r0,8q : Xt ě ε(

Y t8u˘

, (6.23)


ρYε “ inf`

t P r0,8q : supsPr0,ts Ys ě ε(

Y t8u˘

. (6.24)

Then observe that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stopping times τ : Ω Ñr0,8q it holds that

εP`

suptPr0,mintτ,nusXt ě ε˘

“ εP´

D t P r0,mintτ, nus : Xt ě ε¯

“ εP´!

D t P r0,mintτ, nus : Xt ě ε)

X

!

ρXε ď mintτ, nu)¯

“ εP´!

D t P r0,mintτ, nus : Xt ě ε)

X

!

ρXε ď mintτ, nu)

X

!

Xmintτ,n,ρXε uě ε

)¯

ď εP´

Xmintτ,n,ρXε uě ε

¯

“ E

”

ε1tXmintτ,n,ρXε u

ěεu

ı

ď E

”

Xmintτ,n,ρXε u1tX

mintτ,n,ρXε uěεu

ı

ď E“

Xmintτ,n,ρXε u

‰

.

(6.25)

Combining this with the fact for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that mintτ, n, ρXε u is a bounded pFtqtPr0,8q-stopping time(see Exercise 6.2.9) ensures that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that

εP`


ď E“

Xmintτ,n,ρXε u

‰

ď E“

suptPr0,mintτ,n,ρXε usYt‰

ď E“

suptPr0,τ s Yt‰

.(6.26)

Hence, we obtain that for all ε P r0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ

r0,8q it holds that

εP`


“ εP`

YnPN

suptPr0,mintτ,nusXt ě ε(˘

“ ε limnÑ8

P`


ď E“

suptPr0,τ s Yt‰

.(6.27)

This proves (6.18). In the next step we observe that (6.18) ensures that for allε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`


“ P`

suptPr0,τ sXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ˘

“ P´

suptPr0,mintτ,ρYδ usXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ

¯

ď P´

suptPr0,mintτ,ρYδ usXt ě ε

¯

ď 1εE

”

suptPr0,mintτ,ρYδ usYt

ı

“ 1εE

”

min

δ, suptPr0,mintτ,ρYδ usYt(

ı

ď 1εE“

min


.

(6.28)


This proves (6.19). Furthermore, we observe that (6.19) shows that for all ε, δ Pp0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`


ď P`


` P`


ď 1εE“

min


` P`


.

(6.29)

This proves (6.20). Next we note that (6.20) and the Markov inequality show thatfor all r, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that

P`

suptPr0,τ sXt ě r˘

ď 1rE“

min


` P`

min

δ, suptPr0,τ s Yt(

ě δ˘

ď“

1r` 1

δ

‰

E“

min


.

(6.30)

This implies that for all ε, δ, r P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ

p0,8q it holds that

E“

min

ε, suptPr0,τ sXt

(‰

“ E

”

min

ε, suptPr0,τ sXt

(

1tsuptPr0,τsXtăru

ı

`E

”

min

ε, suptPr0,τ sXt

(

1tsuptPr0,τsXtěru

ı

ď mintε, ru ` εP`

suptPr0,τ sXt ě r˘

ď mintε, ru ` ε“

1r` 1

δ

‰

E“

min


ď r ` ε“

1r` 1

δ

‰

E“

min


.

(6.31)

Hence, we obtain that for all ε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñp0,8q it holds that

E“

min

ε, suptPr0,τ sXt

(‰

ď infrPp0,8q

`

r ` εrE“

min


` εδE“

min

δ, suptPr0,τ s Yt(‰˘

ďˇ

ˇεE“

min


ˇ

12

`?εˇ

ˇE“

min


ˇ

12` ε

δE“

min


.

(6.32)

This proves (6.21). Moreover, we note that (6.22) is an immediate consequence of(6.21). The proof of Proposition 6.2.10 is thus completed.


Exercise 6.2.11 (Characterization of convergence in probability). Let pΩ,F ,Pq bea probability space, let pE, dEq be a metric space, and let Xn : Ω Ñ E, n P N0, beF/pE, dEq-strongly measurable mappings. Then the following three statements areequivalent:

(i) For all c P p0,8q it holds that

limnÑ8

E“

mintc, dEpX0, Xnqu‰

“ 0. (6.33)

(ii) There exists a c P p0,8q such that

limnÑ8

E“

mintc, dEpX0, Xnqu‰

“ 0. (6.34)

(iii) For all ε P p0,8q it holds that

limnÑ8

PpdEpX0, Xnq ą εq “ 0. (6.35)

6.2.3 Modifications and indistinguishability

This material is an extended version from Barth et al. 2014. Next we address twonotions that somehow describe when two stochastic processes are “equal up to setsof measure zero”.

Definition 6.2.12 (Modifications). Let pΩ,F ,Pq be a probability space, let pS,Sqbe a measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S be stochasticprocesses such that for every t P T it holds that there exists an event A P F withPpAq “ 1 and

A Ď tXt “ Ytu. (6.36)

Then X and Y are called modifications of each other (i.e., X is called a modificationof Y and Y is called a modification of X).

Exercise 6.2.13. Prove or disprove the following statement: For all measurablespaces pΩ,Fq it holds that tpω, ωq P Ω2 : ω P Ωu P F b F .

Exercise 6.2.14. Specify explicitly measurable spaces pΩ,Fq and pS,Sq and F/S-measurable mappings X, Y : Ω Ñ S such that tX “ Y u “ tω P Ω: Xpωq “ Y pωqu RF . Prove that your result is correct.


Definition 6.2.15 (Indistinguishablility). Let pΩ,F ,Pq be a probability space, letpS,Sq be a measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S bestochastic processes with the property that there exists an event A P F with PpAq “ 1and

A Ď pXtPTtXt “ Ytuq . (6.37)

Then X and Y are called indistinguishable from each other (i.e., X is called indis-tinguishable from Y and Y is called indistinguishable from X).

Let us illustrate Definitions 6.2.12 and 6.2.15 through a simple example (see, e.g.,Kuhn [20]).

Example 6.2.16. Let pΩ,F ,Pq be a probability space, let pFtqtPr0,1s be the filtrationon pΩ,Fq with the property that for all t P r0, 1s it holds that Ft “ F , let τ : Ω Ñ r0, 1sbe an F/Bpr0, 1sq-measurable mapping such that τpPq “ Borelr0,1s, let X, Y : r0, 1s ˆΩ Ñ R be the functions with the property that for all ω P Ω, t P r0, 1s it holds that

Xtpωq “ 0 and Ytpωq “

#

1 : t “ τpωq

0 : t ‰ τpωq. (6.38)

Then

(i) it holds that X, Y are pFtqtPr0,T s-predictable stochastic processes (indeed, letY n : r0, T s ˆ Ω Ñ R, n P N, be the mappings with the property that for alln P N, t P r0, T s it holds that Y n

t pωq “ 1pτpωq´1n,τpωqsptq, observe that forall n P N it holds that Y n is pFtqtPr0,T s-predictable and note that @ pt, ωq Pr0, T s ˆ Ω: limnÑ8 Y

nt pωq “ Ytpωq),

(ii) it holds that τ is an pFtqtPr0,T s-stopping time,

(iii) it holds for all ω P Ω that Xτpωqpωq “ 0 ‰ 1 “ Yτpωqpωq,

(iv) it holds that!

ω P Ω:`

@ t P r0, T s : Xtpωq “ Ytpωq˘

)

“

!

ω P Ω:`

@ t P r0, T s : Ytpωq “ 0˘

)

“ H,(6.39)

(v) it holds for all t P r0, T s that

PpXt “ Ytq “ PpYt “ 0q “ Ppτ ‰ tq “ 1, (6.40)

(vi) and it holds that X and Y are modification of each other but X and Y are notindistinguishable from of each other.


6.2.4 Predictability

Definition 6.2.17 (Predictable sigma-algebra). Let T P r0,8q and let pΩ,Fq be ameasurable space with a filtration pFtqtPr0,T s. Then we denote by Pred

`

pFtqtPr0,T s˘

thesigma-algebra given by

Pred`

pFtqtPr0,T s˘

“

σr0,T sˆΩ

´

tps, ts ˆ A : A P Fs and s, t P r0, T s with s ă tu Y tt0u ˆ A : A P F0u

¯

(6.41)

and we call Pred`

pFtqtPr0,T s˘

the predictable sigma-algebra of pFtqtPr0,T s.

Note, in the setting of Definition 6.2.17, that the definition of the sigma-algebraPred

`

pFtqtPr0,T s˘

depends on the filtration pFtqtPr0,T s.

Definition 6.2.18 (Predictability). Let T P r0,8q, let pS,Sq be a measurable space,let pΩ,Fq be a measurable space with a filtration pFtqtPr0,T s, and let X : r0, T s ˆΩ ÑS be an Pred

`

pFtqtPr0,T s˘

/S-measurable mapping. Then X is called an pFtqtPr0,T s-predictable (stochastic) process (Then X is called pFtqtPr0,T s-predictable).

Observe that if T P p0,8q and if pΩ,Fq is a measurable space with a filtrationpFtqtPr0,T s, then

PredppFtqtPr0,T sq Ď σr0,T sˆΩ

`

B ˆ A : B P Bpr0, T sq and A P FT˘

“ Bpr0, T sq b FT .(6.42)

This is fact is used in the next definition.

Definition 6.2.19 (Product measure on the predictable sigma-algebra). Let T P

p0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s. Then wedenote by

PP,pFtqtPr0,T s : PredpP, pFtqtPr0,T sq Ñ r0,8q (6.43)

the measure given by

PP,pFtqtPr0,T s “ pBorelr0,T sbPq|PredppFtqtPr0,T sq. (6.44)

Let T P p0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s.Then we note that for all t1, t2 P r0, T s, A P Ft1 with t1 ă t2 it holds that

PP,pFtqtPr0,T sppt1, t2s ˆ Aq “ pt2 ´ t1q ¨ PpAq. (6.45)


6.2.5 Construction of the stochastic integral

In the next step we introduce the notion of an elementary process. For this we recallthe notion of a simple function; see Definition 2.3.1 above.

Definition 6.2.20 (Elementary process). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq andpU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let pΩ,Fq be a measurable space with a filtrationpFtqtPr0,T s, and let X : r0, T s ˆ Ω Ñ LpU,Hq be a mapping with the property thatthere exist n P N, 0 ď t1 ă . . . ă tn ď T and for every k P t1, . . . , n ´ 1u anFtk/BpLpU,Hqq-simple function Yk : Ω Ñ LpU,Hq such that for all t P r0, T s it holdsthat

Xt “

n´1ÿ

k“1

Yk ¨ 1ptk,tk`1sptq . (6.46)

Then X is called pFtqtPr0,T s-elementary (or just elementary).

Elementary processes in the sense of Definition 6.2.20 are predictable in the senseof Definition 6.2.18. Let pU, 〈¨, ¨〉U , Üq be an R-Hilbert space, let T P r0,8q, letpΩ,F ,Pq be a probability space, let Q P L1pUq be nonnegative and symmetric, andlet W : r0, T sˆΩ Ñ U be a standard Q-Wiener process. In the stochastic integrationtheory with respect to the possibly infinite dimensional standard Q-Wiener processW the Hilbert space

´

Q12pUq,

⟨Q´

12p¨q, Q´

12p¨q⟩U,›

›Q´12p¨q›

›

2

U

¯

(6.47)

plays an important role. Recall that Q12 is defined according to Theorem 3.4.12and Q´12 is the pseudo inverse of Q12 (see Definition 6.1.5 above). According toProposition 6.1.8 the triple (6.47) is indeed an R-Hilbert space. Lemma 6.2.23 belowillustrate the appearence of the Hilbert space in (6.47) in the stochastic integrationtheory. Before we present Lemma 6.2.23, we note the following exercise and itspreceding remark.

Exercise 6.2.21 (Embedding of LpU,Hq intoHSpQ12pUq, Hq). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , Üq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Prove then that A|Q12pUq P HSpQ12pUq, Hq and that

›

›A|Q12pUq

›

›

HSpQ12pUq,Hq“›

›AQ12›

›

HSpU,Hqď ALpU,Hq

›

›Q12›

›

HSpUqă 8. (6.48)


Remark 6.2.22 (Embedding of LpU,Hq into HSpQ12pUq, Hq). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Then we often simply write A as an abbreviation forA|Q12pUq.

We are now ready to present Lemma 6.2.23, which illustrates the appearence ofthe Hilbert space in (6.47) in the stochastic integration theory.

Lemma 6.2.23. Let T P r0,8q, s, t P r0, T s with s ď t, let pΩ,F ,P, pFtqtPr0,T q bea stochastic basis, let pU, 〈¨, ¨〉U , ¨Uq be a separable R-Hilbert space, let A : Ω Ñ

LpU,Hq be an Fs/BpLpU,Hqq-simple function, let Q P L1pUq be nonnegative andsymmetric, and let W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t.pFtqtPr0,T s. Then

E“

A pWt ´Wsq2H

‰

“ E

”

A2HSpQ12pUq,Hq

ı

pt´ sq . (6.49)

Proof of Lemma 6.2.23. Let B Ď U be an orthonormal basis of U and let λ : B Ñr0,8q be a globally bounded function such that for all u P U it holds that

Qu “ÿ

bPB

λb 〈b, u〉U b. (6.50)

Then note that the fact that B Ď U is an orthonormal basis of U and the continuityof A imply that

E“

A pWt ´Wsq2H

‰

“ E

»

–

›

›

›

›

›

A

˜

ÿ

bPB

〈b,Wt ´Ws〉U b

¸›

›

›

›

›

2

H

fi

fl

“ E

»

–

›

›

›

›

›

ÿ

bPB

〈b,Wt ´Ws〉U Ab

›

›

›

›

›

2

H

fi

fl

“ E

«

ÿ

b1,b2PB

〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉H

ff

“ÿ

b1,b2PB

Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉Hs .

(6.51)


Independency and the definition of a standard Q-Wiener process hence implies that

E“

A pWt ´Wsq2H

‰

“ÿ

b1,b2PB

Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U sEr〈Ab1, Ab2〉Hs

“ÿ

bPB

E“

|〈b,Wt ´Ws〉U |2‰

E“

Ab2H‰

“ pt´ sqÿ

bPB

〈b,Qb〉U E“

Ab2H‰

“ pt´ sqÿ

bPλ´1pp0,8qq

〈b,Qb〉Ulooomooon

“λb

E“

Ab2H‰

.

(6.52)

This proves that

E“

A pWt ´Wsq2H

‰

“ pt´ sqE

»

–

ÿ

bPλ´1pp0,8qq

›

›

›Aa

λb b›

›

›

2

H

fi

fl

“ pt´ sqE

»

–

ÿ

bPλ´1pp0,8qq

›

›AQ12b›

›

2

H

fi

fl .

(6.53)

Next we observe that the set

Q12pbq P im

`

Q12˘

: b P λ´1pp0,8qq

(

(6.54)

is an orthonormal basis in the R-Hilbert space´

Q12pUq,

⟨Q´

12p¨q, Q´

12p¨q⟩U,›

›Q´12p¨q›

›

2

U

¯

. (6.55)

Combining this with (6.53) completes the proof of Lemma 6.2.23.

Lemma 6.2.24 (Ito’s isometry in infinite dimension for elementary processes). Letn P t2, 3, . . . u, T P r0,8q, 0 ď t1 ă ¨ ¨ ¨ ă tn “ T , let pΩ,F ,P, pFtqtPr0,T q bea stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, for every k P t1, 2, . . . , n ´ 1u let Yk : Ω Ñ LpU,Hq be an Ftk/BpLpU,Hqq-simple function, let Q P L1pUq be nonnegative and symmetric, and let W : r0, T s ˆΩ Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s. Then

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl “ E

«

n´1ÿ

k“1

Yk2HSpQ12pUq,Hq ptk`1 ´ tkq

ff

“

ż T

0

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk 1ptk,tk`1spsq

›

›

›

›

›

2

HSpQ12pUq,Hq

fi

fl ds.

(6.56)


Proof of Lemma 6.2.24. Note that

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl

“

n´1ÿ

k,l“1

E“⟨Yk

`

Wtk`1´Wtk

˘

, Yl`

Wtl`1´Wtl

˘⟩H

‰

“ 2ÿ

k,lPt1,...,n´1ukăl

E“⟨rYls

˚Yk`

Wtk`1´Wtk

˘

,Wtl`1´Wtl

⟩U

‰

`

n´1ÿ

k“1

E

”

›

›Yk`

Wtk`1´Wtk

˘›

›

2

H

ı

.

(6.57)

Independency and Lemma 6.2.23 hence imply that for all orthonormal bases B Ď Uof U it holds that

E

»

–

›

›

›

›

›

n´1ÿ

k“1

Yk`

Wtk`1´Wtk

˘

›

›

›

›

›

2

H

fi

fl

“ 2ÿ


E

«

ÿ

bPB

⟨b, rYls

˚Yk`

Wtk`1´Wtk

˘⟩U

⟨b,Wtl`1

´Wtl

⟩U

ff

`

n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq

“ 2ÿ


ÿ

bPB

E“⟨b, rYls

˚Yk`

Wtk`1´Wtk

˘⟩U

‰

E“⟨b,Wtl`1

´Wtl

⟩U

‰

`

n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq “n´1ÿ

k“1

E

”

Yk2HSpQ12pUq,Hq

ı

ptk`1 ´ tkq .

(6.58)


The random variableřn´1k“1 Yk

`

Wtk`1´Wtk

˘

in (6.56) will be the stochastic in-

tegral of the elementary stochastic processřn´1k“1 Yk 1ptk,tk`1s in (6.56). Our aim is to

integrate more general stochastic processes. To do so the following lemma is crucial.


Lemma 6.2.25 (Density). Let T P r0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let Q P L1pUqbe nonnegative and symmetric, and let X P L0

`

PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq

˘

satisfy

that it holds P-a.s. thatşT

0Xs

2HSpQ12pUq,Hq

ds ă 8. Then there exists a sequence

Xn : r0, T s ˆΩ Ñ LpU,Hq, n P N, of pFtqtPr0,T s-elementary stochastic processes suchthat for all ε P p0,8q it holds that

limnÑ8

Pˆż T

0

Xs ´Xns

2HSpQ12pUq,Hqq ds ě ε

˙

“ 0. (6.59)

Lemma 6.2.25 follows, e.g., from Proposition 2.3.8 in Prevot & Rockner [24].In the next result, Theorem 6.2.26, the existence and uniqueness of the stochasticintegral is established (cf., e.g., Proposition 2.26 in Karatzas & Shreve [16]).

Theorem 6.2.26 (Stochastic integral). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq beseparable R-Hilbert spaces, let T P p0,8q, let Q P L1pUq be nonnegative and sym-metric, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, and let W : r0, T s ˆΩ Ñ U be astandard Q-Wiener process w.r.t. pFtqtPr0,T s. Then there exists a unique linear map-

ping I :

X P L0`


˘

: P`

∫T0 Xs2HSpQ12pUq,Hq

ds ă 8˘

“

1(

Ñ L0pP ; ¨Hq which satisfies

(i) that for all Xn P DpIq, n P N, with limnÑ8E“

mint1,şT

0Xn

s 2HSpQ12pUq,Hq

dsu‰

“

0 it holds that limnÑ8E“

mint1, IpXnqHu‰

“ 0 (continuity) and

(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds P -a.s. that

Ip1ps,tsXq “ X pWt ´Wsq (6.60)

(stochastic integration of elementary stochastic processes).


Definition 6.2.27 (Stochastic integral on the entire time interval). Let T P p0,8q,let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üqbe separable R-Hilbert spaces, let Q P L1pUq be nonnegative and symmetric, andlet W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s. Then we

denote by IW :

X P L0`


˘

: P`

∫T0 Xs2HSpQ12pUq,Hq

ds ă

8˘

“ 1(

Ñ L0pP ; ¨Hq the unique linear mapping which satisfies

(i) that for all Xn P DpIW q, n P N, with limnÑ8E“

mint1,şT

0Xn

s 2HSpQ12pUq,Hq

dsu‰

“ 0 it holds that limnÑ8E“

mint1, IW pXnqHu‰

“ 0 (continuity) and

(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds P -a.s. that

IW p1ps,tsXq “ X pWt ´Wsq (6.61)

(stochastic integration of elementary stochastic processes).

Question 6.2.28. Let T P p0,8q, a, b P r0, T s with a ă b, let pΩ,F , P, pFtqtPr0,T sqbe a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbertspaces, let Q P L1pUq be nonnegative and symmetric, let W : r0, T sˆΩ Ñ U be a stan-dard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P L0

`


˘

.

Is it true or is it not true that r0, T s ˆ Ω ÞÑ 1pa,bqpsq ¨ Xspωq P HSpQ12pUq, Hqq P

L0`


˘

?

Definition 6.2.29 (Stochastic integral). Let T P p0,8q, a, b P r0, T s with a ă b,let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üqbe separable R-Hilbert spaces, let Q P L1pUq be nonnegative and symmetric, letW : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P

L0`


˘

satisfy that it holds P-a.s. that ∫ ba Xs2HSpQ12pUq,Hq

ds

ă 8. Then we denote byşb

aXs dWs P L

0pP; ¨Hq the element given byşb

aXs dWs “

IW p1pa,bqXq.

Exercise 6.2.30. Let T P r0,8q, a, b P r0, T s with a ă b, let pΩ,F ,P, pFtqtPr0,T sqbe a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbertspaces, let Q P L1pUq be nonnegative and symmetric, let W : r0, T sˆΩ Ñ U be a stan-dard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P L0

`

PpP,pFtqtPr0,T sq; ¨HSpQ12pUq,Hq

˘

be such thatşb

aE“

Xs2HSpQ12pUq,Hq

‰

ds ă 8. Prove thatşb

aE“

Xs2H

‰

ds ă 8 and

E“ şb

aXs dWs

‰

“ 0.


In the following a few properties of the stochastic integral are collected. For thisthe following lemma is used.

Lemma 6.2.31. Let T P r0,8q, t P r0, T s, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pS,Sq be a measurable space, let X : Ω Ñ S be an F/S-measurable mappingand let Y : Ω Ñ S be an Ft/S-measurable mapping such that it holds P-a.s. thatX “ Y . Then it holds that X is Ft/S-measurable.

Proof of Lemma 6.2.31. First, note that the assumption that X “ Y P-a.s. showsthat there exists a measurable set A P F with the property that PpAq “ 1 and withthe property that for all ω P A it holds that Xpωq “ Y pωq. Next observe that for allB P S it holds that

X´1pBq “

“

X´1pBq X A

‰

Y“

X´1pBqzA

‰

“ tω P A : Xpωq P Bu Y“

X´1pBqzA

‰

“ tω P A : Y pωq P Bu Y“

X´1pBqzA

‰

““

Y ´1pBq X A

‰

Y“

X´1pBqzA

‰

.

(6.62)

Moreover, observe that the assumption that pFtqtPr0,T s is a normal filtration togetherwith the fact that PpAq “ 1 implies that

A,Ac P F0 Ď Ft Ď F . (6.63)

This and the assumption that Y is Ft/S-measurable prove that for all B P S it holdsthat

Y ´1pBq X A P Ft. (6.64)

Furthemore, note that the monotonicity of the probability measure P ensures thatfor all B P S it holds that PpX´1pBqzAq “ 0. The assumption that pFtqtPr0,T s isnormal hence shows that for all B P S it holds that

X´1pBqzA P Ft. (6.65)

Combining (6.62) with (6.64) and (6.65) proves that for all B P S it holds thatX´1pBq P Ft. The proof of Lemma 6.2.31 is thus completed.

Consider the setting of Lemma 6.2.31 and let pV, ¨V q be a separable Banachspace. Then Lemma 6.2.31, in particular, proves that for all t1, t2 P r0, T s witht1 ď t2 it holds that

L0`

P|Ft1 ; ¨V˘

Ď L0`

P|Ft2 ; ¨V˘

Ď L0`

P; ¨V˘

. (6.66)


In Lemma 6.2.31 it is crucial that the filtration is normal. Let us collect a few prop-erties of the stochastic integral with possibly infinite dimensional Wiener processesas integrator processes.


Theorem 6.2.32 (Properties of the stochastic integral). Let T P p0,8q, a, b P r0, T swith a ă b, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbert spaces,let Q P L1pUq be nonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s, andlet X : r0, T sˆΩ Ñ HSpQ12pUq, Hq be an pFtqtPr0,T s-predictable stochastic processes

with the property that it holds P-a.s. thatşb

aXs

2HSpQ12pUq,Hq ds ă 8. Then

(i) it holds thatşb

aXs dWs P L

0`

P|Fb ; ¨H˘

, i.e., it holds thatşb

aXs dWs is Fb/BpHq-

measurable,

(ii) it holds that pşt

aXs dWsqtPra,bs is an pFtqtPra,bs-adapted stochastic process,

(iii) for all α, β P R and all pFtqtPr0,T s-predictable stochastic processes Y, Z : r0, T sˆ

Ω Ñ HSpQ12pUq, Hq with P` şb

aYs

2HSpQ12pUq,Hq

`Zs2HSpQ12pUq,Hq

ds ă 8˘

“

1 it holds P-a.s. thatż b

a

rαYs ` βZss dWs “ α

ż b

a

Ys dWs ` β

ż b

a

Zs dWs, (6.67)

(iv) for all pFtqtPr0,T s-predictable stochastic processes Y : r0, T sˆΩ Ñ HSpQ12pUq, Hq

withşb

aE“

Ys2HSpQ12pUq,Hq

‰

ds ă 8 it holds that

E

«

›

›

›

›

ż b

a

Ys dWs

›

›

›

›

2

H

ff

“

ż b

a

E

”

Ys2HSpQ12pUq,Hq

ı

ds ă 8, (Ito’s isometry)

›

›

›

›

ż b

a

Ys dWs

›

›

›

›

L2pP;¨Hq

“

ˆż b

a

Ys2L2pP;¨

HSpQ12pUq,Hqqds

˙

12

ă 8, (6.68)

E

„ż b

a

Ys dWs

“ 0, (6.69)

(v) for all p P r2,8q it holds that

›

›

›

›

ż b

a

Xs dWs

›

›

›

›

LppP;¨Hq

ď

c

p pp´ 1q

2

ˆż b

a

Xs2LppP;¨

HSpQ12pUq,Hqqds

˙

12

˜

E

«

›

›

›

›

ż b

a

Xs dWs

›

›

›

›

p

H

ff¸1p

ď

c

p pp´ 1q

2

ˆż b

a

´

E

”

Xsp

HSpQ12pUq,Hq

ı¯2pds

˙

12

,

(Burkholder-Davis-Gundy inequality I)


(vi) there exists an up to indistinguishability unique pFtqtPra,bs-adapted stochasticprocess V : ra, bs ˆ Ω Ñ H with continuous sample paths which satisfies thatfor all t P ra, bs it holds P -a.s. that Vt “

şt

aXs dWs (V is called a continuous

modification of pşt

aXs dWsqtPra,bs),

(vii) and for all continuous modifications V : ra, bsˆΩ Ñ H of pşt

aXs dWsqtPra,bs and

all p P r2,8q it holds that›

›

›

›

›

supsPra,bs

VsH

›

›

›

›

›

LppP;|¨|Rq

ď p

ˆż b

a

Xs2LppP;¨

HSpQ12pUq,Hqqds

˙

12

,

˜

E

«

supsPra,bs

VspH

ff¸1p

ď p

ˆż b

a

´

E

”

Xsp

HSpQ12pUq,Hq

ı¯2p

ds

˙

12

.

(Burkholder-Davis-Gundy inequality II)

The statements of Theorem 6.2.32 and their proofs can, for example, be found in[24] and [7].

Exercise 6.2.33 (Stochastic integration of L2-continuous stochastic processes). LetT P p0,8q, d,m P N, let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let W : r0, T sˆΩ ÑRm be an m-dimensional standard pFtqtPr0,T s-Brownian motion, let a, b P r0, T s witha ď b and let X : r0, T s ˆ Ω Ñ Rdˆm be an pFtqtPr0,T s-predictable stochastic processwith X P Cpr0, T s, L2pP ; ¨

Rdˆmqq. Prove then that

ż b

a

Xs dWs “ L2pP ; ¨

Rdq´ lim

nÑ8

«

n´1ÿ

k“0

Xpa` kpbáq

nq

´

Wa` pk`1qpbáq

n´W

a` kpbáqn

¯

ff

.

(6.70)

6.2.6 Elementary processes revisited

In the literature (see, e.g., Definition 2.3.1 in [24]) a slightly different notion ofan elementary stochastic process is often given. In Proposition 6.2.37 below weshow that these different definitions (cf. Definition 2.3.1 in [24] and Definition 6.2.20above) are equivalent. Our proof of Proposition 6.2.37 uses Exercise 6.2.35 below.Exercise 6.2.35 below can, e.g., be proved by using the following lemma.

Lemma 6.2.34. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space andlet ϕ P CpH,Kq, v P Hzt0u with the property that for all u P H it holds thatϕpuq ¨ 〈v, u〉H “ 0. Then it holds for all u P H that ϕpuq “ 0.


Proof of Lemma 6.2.34. Observe that for all λ P Kzt0u, w P rspantvusJ it holds that

0 “ ϕpλv ` wq ¨ 〈v, λv ` w〉H “ ϕpλv ` wq ¨ λ ¨ v2Hlooomooon

‰0

. (6.71)

This implies that for all for all λ P Kzt0u, w P rspantvusJ it holds that

ϕpλv ` wq “ 0. (6.72)

The fact that the set!

λv ` w P H : λ P Kzt0u, w P rspantvusJ)

(6.73)

is dense in H together with the assumption that ϕ is continuous hence implies that forall u P H it holds that ϕpuq “ 0. The proof of Lemma 6.2.34 is thus completed.

Exercise 6.2.35. Let K P tR,Cu, n P t2, 3, . . . u, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2ube K-Hilbert spaces and let A1, . . . , An P LpH1, H2q with the property that for all u PH1 it holds that A1u P tA2u,A3u, . . . , Anuu. Prove then that A1 P tA2, A3, . . . , Anu.

Exercise 6.2.35 can, e.g., be proved by using Lemma 6.2.34. In our proof ofProposition 6.2.37 below we also use the following exercise.

Exercise 6.2.36. Let pΩ1,F1q be a measurable space, let Ω2 be a set and let f : Ω1 Ñ

Ω2 be a mapping with the property that the set impfq is finite. Then f is F1/PpΩ2q-measurable if and only if for all ω P impfq it holds that f´1ptωuq P F1.

Proposition 6.2.37 (Uniform and strong measurability). Let K P tR,Cu, let pΩ,Fqbe a measurable space, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces and letY : Ω Ñ LpH1, H2q be a function with the property that impY q is a finite set. Thenit holds that Y is F/BpLpH1, H2qq-measurable if and only if for all v P H1 it holdsthat Ω Q ω ÞÑ Y pωqv P H2 is F/BpH2q-measurable.

Proof of Proposition 6.2.37. It is clear that if Y is F/BpLpH1, H2qq-measurable, thenit holds for all v P H1 that Y v is F/BpH2q-measurable. We thus assume in thefollowing that for all v P H1 it holds that Y v is F/BpH2q-measurable. Let A1 P impY qbe arbitrary. We now prove that Y ´1ptA1uq P F . This and Exercise 6.2.36 will thenshow that Y is F/BpLpH1, H2qq-measurable. W.l.o.g. we assume that impY qztA1u ‰

H. As impY q is a finite set, there exists n P t2, 3, . . . u and A2, . . . , An P impY qztA1u

such thattA1, A2, . . . , Anu “ impY q. (6.74)


Exercise 6.2.35 implies that there exists a vector u P H1 such that

A1u R tA2u, . . . , Anuu . (6.75)

This and the assumption that Y u is F/BpH2q-measurable imply that

F Q pY uq´1ptA1uuq “ Y ´1

ptA1uq. (6.76)

Exercise 6.2.36 thus completes the proof of Proposition 6.2.37.

Exercise 6.2.38. Let pΩ,Fq be a measurable space, let pE, dEq be a metric spaceand let f : Ω Ñ E be a mapping with the property that the set impfq is finite. Thenf is F/PpEq-measurable if and only if f is F/BpEq-measurable.

6.2.7 Cylindrical Wiener process

The following presentations are based on [24] and Chapter 5 in [14]. Let pU, 〈¨, ¨〉U , ¨Uqbe an R-Hilbert space and let Q P L1pUq be nonnegative and symmetric. In Subsec-tion 6.1.1 the notion of a standard Q-Wiener process is presented. The covarianceoperator Q associated to a standard Q-Wiener process is a nonnegative, symmetric,and nuclear linear operator on the Hilbert space U on which the standard Wienerprocess takes values in. In many situations one is interested in an infinite dimen-sional Wiener process with a covariance operator that is a nonnegative and symmetricbounded linear operator which is not a nuclear linear operator (such as, for example,the identity operator on an infinite dimensional Hilbert space). This can be achievedby the concept of a cylindrical Wiener process.

Definition 6.2.39 (Cylindrical Wiener process). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hqand pH1, 〈¨, ¨〉H1

, ¨H1q be R-Hilbert spaces with H Ď H1 continuously, let Q P LpHq

and Q1 P L1pH1q be nonnegative and symmetric with the property that

`

Q12pHq,

›

›Q´12p¨q›

›

H

˘

“`

Q12

1 pH1q,›

›Q´12

1 p¨q›

›

H1

˘

, (6.77)

let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space, and let W : r0, T s ˆ Ω Ñ H1

be a standard Q1-Wiener process w.r.t. pFtqtPr0,T s. Then W is called a cylindricalQ-Wiener process w.r.t. pFtqtPr0,T s.

Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let Q P LpHq benonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space,and let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s. The cylindricalQ-Wiener process pWtqtPr0,T s thus, in general, does not take values in the Hilbert space


H, on which the covariance operator Q associated to W is defined, but on a largerHilbert space with a weaker topology into which H is continuously embedded. Moreresults on cylindrical Q-Wiener processes can be found in Section 2.5 in Prevot &Rockner [24].

Part III

Stochastic Partial DifferentialEquations (SPDEs)

179

Chapter 7

Solutions of SPDEs

7.1 Existence, uniqueness and properties of mild

solutions of SPDEs

7.1.1 Mild solutions of SPDEs

The next definition presents what we mean by a stochastic partial differential equa-tion and a mild solution of it.

181

182 CHAPTER 7. SOLUTIONS OF SPDES

Definition 7.1.1 (Mild solutions). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be sep-arable R-Hilbert spaces, let Q P LpUq be nonnegative and symmetric, let A : DpAq ĎH Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 8, let η PpsuppσP pAqq,8q, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces

associated to η ´ A, let T P r0,8q, α, β, γ P R, O P BpHγq, F PM`

BpOq,BpHαq˘

,

B PM`

BpOq,BpHSpQ12pUq, Hβqq˘

, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, letξ PMpF0,BpOqq, let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s,and let X : r0, T s ˆ Ω Ñ O be an pFtqtPr0,T s-predictable stochastic process with theproperty that for all t P r0, T s it holds P-a.s. that

ż t

0

eApt´sqF pXsqHγ ` eApt´sqBpXsq

2HSpQ12pUq,Hγq

ds ă 8 (7.1)

and Xt “ eAtξ `

ż t

0

eApt´sqF pXsq ds`

ż t

0

eApt´sqBpXsq dWs. (7.2)

Then X is said to be a mild solution of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.3)

Equation (7.3) is referred to as stochastic partial differential equation (SPDE), thefunction F is the nonlinear part of the drift coefficient function Av ` F pvq, v P O,and the function B is the diffusion coefficient function of the SPDE (7.3).

7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 183

Example 7.1.2 (Ornstein-Uhlenbeck processes (stochastic heat equation with ad-ditive noise)). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplaceoperator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq (see Definition 3.5.7),let ξ P L2pBorelp0,1q; |¨|Rq, let en P L2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all

n P N and Borelp0,1q-almost all x P p0, 1q it holds that enpxq “?

2 sinpnπxq, letpΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let rn P r0,8q, n P N, be a bounded se-quence of nonnegative real numbers, let Q : L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rqbe the mapping with the property that for all v P L2pBorelp0,1q; |¨|Rq it holds that

Qv “8ÿ

n“1

rn 〈en, v〉L2pBorelp0,1q;|¨|Rqen, (7.4)

let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T s ˆΩ Ñ H be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t Pr0, T s it holds P-a.s. that

Xt “ eAtξ `

ż t

0

eApt´sq dWs. (7.5)

Then X is a mild solution of the SPDE

dXt “ AXt dt` dWt, t P r0, T s, X0 “ ξ. (7.6)

Sometimes one also writes

dXtpxq “B2

Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq (7.7)

for t P r0, T s, x P p0, 1q as a short form for (7.6).

In the following we investigate a few further properties of mild solutions of SPDEs.To this end we frequently use the following setting.

7.1.2 A setting for SPDEs with globally Lipschitz continuousnonlinearities

Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be two separable R-Hilbert spaces, letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associatedto ´A, let T P r0,8q, p P r2,8q, γ P R, η P r0, 1q, β P rγ ´ η2, γs, F P

C0,1pHγ, Hγ´ηq, B P C0,1pHγ, HSpU,Hβqq, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let ξ P LppP|F0 ; ¨Hγ q, and let pWtqtPr0,T s be a cylindrical IdU -Wiener process

w.r.t. pFtqtPr0,T s.


7.1.3 A strong perturbation estimate for SPDEs

The following estimate is a special case of an inequality known as Minkowski’s integralinequality; see, for instance, Apendix A.1 in Stein [26] and Theorem 202 in Hardy,Littlewood & Polya [9].

Proposition 7.1.3. Let T P r0,8q, p P r1,8q, let pΩ,F ,Pq be a probability space,and let Y : r0, T s ˆ Ω Ñ r0,8s be a product measurable stochastic process. Then

ˆ

E„ˇ

ˇ

ˇ

ˇ

ż T

0

Ys ds

ˇ

ˇ

ˇ

ˇ

p˙1p

ď

ż T

0

´

E“

|Ys|p‰

¯1pds. (7.8)

In the next result, Proposition 7.1.4, we establish, in the setting of Section 7.1.2,a certain strong perturbation result for two arbitrary predictable stochastic processesX1 and X2 satisfying supsPr0,T s maxkPt1,2u X

ks LppP;¨Hγ q

ă 8. In particular, we em-

phasize in the setting of Section 7.1.2 that neither X1 nor X2 in Proposition 7.1.4need to be mild solutions of the SPDE


In the statement of Proposition 7.1.4 and in a number of other results in this andthe next chapter we use the functions Er : r0,8q Ñ r0,8q, r P p0,8q, introduced inDefinition 1.3.1 in Chapter 1 above.

Proposition 7.1.4 (Perturbation estimate). Assume the setting in Section 7.1.2and let X1, X2 : r0, T s ˆ Ω Ñ Hγ be pFtqtPr0,T s-predictable stochastic processes withthe property that supsPr0,T s maxkPt1,2u X

ks LppP;¨Hγ q

ă 8. Then

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?

2 |F |C0,1pHγ,Hγ´ηq?1´η

`a

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq

¨?

2 suptPr0,T s

›

›

›

›

„

X1t ´

ż t

0

eApt´sqF pX1s q ds´

ż t

0

eApt´sqBpX1s q dWs

`

„ż t

0

eApt´sqF pX2s q ds`

ż t

0

eApt´sqBpX2s q dWs ´X

2t

›

›

›

›

LppP;¨Hγ q

ă 8.

(7.10)


Proof of Proposition 7.1.4. Note that the Minkowski inequality ensures that for allt P r0, T s it holds that

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď

›

›

›

›

›

„

X1t ´

ˆż t

0


ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0


ż t

0

eApt´sqBpX2s q dWs

˙

´X2t

›

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

›

ˆż t

0


ż t

0

eApt´sqBpX1s q dWs

˙

´

ˆż t

0


ż t

0

eApt´sqBpX2s q dWs

˙

›

›

›

›

›

LppP;¨Hγ q

.

(7.11)

Again the Minkowski inequality hence implies that for all t P r0, T s it holds that

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď

›

›

›

›

›

„

X1t ´

ˆż t

0


ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0


ż t

0

eApt´sqBpX2s q dWs

˙

´X2t

›

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

.

(7.12)


Next note that Holder’s inequality implies that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

ď

ż t

0

›

›eApt´sq“

F pX1s q ´ F pX

2s q‰›

›

LppP;¨Hγ qds

ď

ż t

0

F pX1s q ´ F pX

2s qLppP;¨Hγ´η

q

pt´ sqηds

ď |F |C0,1pHγ ,Hγ´ηq

ż t

0

X1s ´X

2s LppP;¨Hγ q

pt´ sqηds


g

f

f

e

ż t

0

pt´ sq´η ds

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds

“ |F |C0,1pHγ ,Hγ´ηq

d

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η X1s ´X

2s

2LppP;¨Hγ q

ds.

(7.13)

Furthermore, observe that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

ď

d

p pp´1q2

ż t

0

eApt´sq rBpX1s q ´BpX

2s qs

2LppP;¨HSpU,Hγ qq

ds

ď

d

p pp´1q2

ż t

0

pt´ sq´η BpX1s q ´BpX

2s q

2LppP;¨HSpU,Hγ´η2q

qds

ď |B|C0,1pHγ ,HSpU,Hγ´η2qq

d

p pp´1q2

ż t

0

pt´ sq´η X1s ´X

2s

2LppP;¨Hγ q

ds.

(7.14)


Combining (7.13) and (7.14) proves that for all t P r0, T s it holds that

›

›

›

›

ż t

0

eApt´sq“

F pX1s q ´ F pX

2s q‰

ds

›

›

›

›

LppP;¨Hγ q

`

›

›

›

›

ż t

0

eApt´sq“

BpX1s q ´BpX

2s q‰

dWs

›

›

›

›

LppP;¨Hγ q

ď

„

|F |C0,1pHγ ,Hγ´ηqtp1´ηq2?

1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq

?p pp´1q?

2

¨

g

f

f

e

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds.

(7.15)

Putting this into (7.12) and using the fact that @ a, b P R : pa`bq2 ď 2a2`2b2 provesthat for all t P r0, T s it holds that

›

›X1t ´X

2t

›

›

2

LppP;¨Hγ q

ď 2

›

›

›

›

›

„

X1t ´

ˆż t

0


ż t

0

eApt´sqBpX1s q dWs

˙

`

„ˆż t

0


ż t

0

eApt´sqBpX2s q dWs

˙

´X2t

›

›

›

›

›

2

LppP;¨Hγ q

`

”

|F |C0,1pHγ ,Hγ´ηq

?2T

p1´ηq2?

1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq

a

p pp´ 1qı2

¨

ż t

0

X1s ´X

2s

2LppP;¨Hγ q

pt´ sqηds.

(7.16)

The generalized Gronwall inequality in Corollary 1.4.6 hence completes the proof ofProposition 7.1.4.

The next corollary, Corollary 7.1.5, is an immediate consequence of the strongperturbation estimate in Proposition 7.1.4 above.


Corollary 7.1.5 (Perturbation in the initial value). Assume the setting in Sec-tion 7.1.2 and let X1, X2 : r0, T s ˆ Ω Ñ Hγ be pFtqtPr0,T s-predictable stochastic pro-cesses with the property that supsPr0,T s maxkPt1,2u X

ks LppP;¨Hγ q

ă 8 and with the

property that for all t P r0, T s, k P t1, 2u it holds P-a.s. that

Xkt “ eAtXk

0 `

ż t

0

eApt´sqF pXks q ds`

ż t

0

eApt´sqBpXks q dWs. (7.17)

Then

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ qď?

2›

›X10 ´X

20

›

›

LppP;¨Hγ q

¨ Ep1´ηq„

T 1´η?


`a


ă 8.(7.18)

7.1.4 Uniqueness of mild solutions of SPDEs

7.1.4.1 Uniqueness of predictable mild solutions of SEEs with globallyLipschitz continuous coefficients

As an immediate consequence of Corollary 7.1.5, we obtain, under suitable assump-tions, uniqueness of mild solutions of SPDEs; cf., e.g., Theorem 7.4 (i) in Da Prato& Zabczyk [7].

Corollary 7.1.6. Assume the setting in Subsection 7.1.2, let X1, X2 : r0, T s ˆ Ω ÑHγ be mild solutions of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (7.19)

and assume that maxkPt1,2u suptPr0,T s Xkt LppP;¨Hγ q

ă 8. Then it holds that X1 and

X2 are modifications of each other, i.e., it holds for all t P r0, T s that P`

X1t “ X2

t

˘

“

1.

7.1.4.2 Uniqueness of left-continuous mild solutions of SEEs with semi-globally Lipschitz continuous coefficients

The proof of the next result, Proposition 7.1.7, is similiar to the proof of Theorem 7.4in Da Prato & Zabczyk [7]. See also, e.g., Lemma 8.2 in [28] for the next result.


Proposition 7.1.7 (Local solutions). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq beseparable R-Hilbert spaces, let A : DpAq Ď H Ñ H be a symmetric diagonal lin-ear operator with suppσP pAqq ă 0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family ofinterpolation spaces associated to ´A, let T P p0,8q, γ P R, η P r0, 1q, F P

CpHγ, Hγ´ηq, B P CpHγ, HSpU,Hγ´η2qq satisfy for all bounded sets E Ď Hγ that|F |E|C0,1pE,Hγ´ηq`|B|E|C0,1pE,HSpU,Hγ´η2qq ă 8, let pΩ,F ,Pq be a probability space witha normal filtration pFtqtPr0,T s, let τk : Ω Ñ r0, T s, k P t1, 2u, be pFtqtPr0,T s-stoppingtimes, and let Xk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s-adapted stochas-tic processes with left-continuous and bounded sample paths and with the propertythat for all k P t1, 2u, t P r0, T s it holds P-a.s. that

şt

01tsăτku

“

ept´sqAF pXks qHγ `

ept´sqABpXks q

2HSpU,Hγq

‰

ds ă 8 and

Xkt 1ttďτku (7.20)

“

„

etAXk0 `

ż t

0

1tsăτku ept´sqAF pXk

s q ds`

ż t

0

1tsăτku ept´sqABpXk

s q dWs

1ttďτku.

Then P`

@ t P r0, T s : 1tX10“X

20uX1

mintt,τ1,τ2u“ 1tX1

0“X20uX2

mintt,τ1,τ2u

˘

“ 1.

Proof of Proposition 7.1.7. Throughout this proof let Ω P F be the set given byΩ “ tX0 “ Y0u, let %r,k : Ω Ñ r0, T s, r P p0,8q, k P t1, 2u, be the mappings with theproperty that for all r P p0,8q, k P t1, 2u it holds that

%r,k “ inf`

tT u Y tt P r0, T s : Xkt Hγ ą ru

˘

, (7.21)

let ρr : Ω Ñ r0, T s, r P p0,8q, be the mappings with the property that for allr P p0,8q it holds that ρr “ mint%r,1, %r,2, τ1, τ2u, and let Xk,r : r0, T s ˆ Ω Ñ Hγ,k P t1, 2u, r P p0,8q, be the mappings with the property that for all k P t1, 2u,r P p0,8q, t P r0, T s it holds that Xk,r

t “ 1ΩXttďρruXkt . Note for all r P p0,8q,

k P N that %r,k and ρr are pFtqtPr0,T s-stopping times. This ensures that for everyr P p0,8q, k P N it holds that Xk,r is a pFtqtPr0,T s-predictable stochastic process withleft-continuous sample paths. Moreover, observe that for all r P p0,8q, t P r0, T s it


holds P-a.s. that

X1,rt ´X2,r

t “ 1ΩXttďρru

ż t

0

ept´sqA“

1tsăτ1uF pX1s q ´ 1tsăτ2uF pX

2s q‰

ds

` 1ΩXttďρru

ż t

0

ept´sqA“

1tsăτ1uBpX1s q ´ 1tsăτ2uBpX

2s q‰

dWs

“ 1ΩXttďρru

ż t

0

1ΩXtsăρru ept´sqA

“

F pX1s q ´ F pX

2s q‰

ds

` 1ΩXttďρru

ż t

0


“

BpX1s q ´BpX

2s q‰

dWs

“ 1ΩXttďρru

ż t

0


“

F pX1,rs q ´ F pX

2,rs q

‰

ds

` 1ΩXttďρru

ż t

0


“

BpX1,rs q ´BpX

2,rs q

‰

dWs.

(7.22)

This implies for all r P p0,8q that

supsPr0,T s

E“

X1,rs ´X2,r

s 2Hγ

‰

“ supsPr0,T s

E“

1ΩXtsďρruX1s ´X

2s

2Hγ

‰

ď supsPr0,T s

E“

1ΩXtsďρruXtρr“0uX1s ´X

2s

2Hγ

‰

` supsPr0,T s

E“

1ΩXtsďρruXtρrą0uX1s ´X

2s

2Hγ

‰

“ supsPr0,T s

E“

1ΩXtsďρruXtρrą0uX1s ´X

2s

2Hγ

‰

ď supsPr0,T s

E“

1ΩXtρrą0uX1mints,ρru ´X

2mints,ρru

2Hγ

‰

ď 2 ¨ supsPr0,T s

E“

1tρrą0uX1mints,ρru

2Hγ ` 1tρrą0uX

2mints,ρru

2Hγ

‰

ď 4r2ă 8.

(7.23)

Moreover, equation (7.22) ensures that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2E

«

ˇ

ˇ

ˇ

ˇ

ż t

0

1ΩXtsăρru›

›ept´sqA“

F pX1,rs q ´ F pX

2,rs q

‰›

›

Hγds

ˇ

ˇ

ˇ

ˇ

2ff

` 2E

«

›

›

›

›

ż t

0


“

BpX1,rs q ´BpX

2,rs q

‰

dWs

›

›

›

›

2

Hγ

ff

.

(7.24)


Ito’s isometry hence proves that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2E

«

ˇ

ˇ

ˇ

ˇ

ż t

0

1ΩXtsăρru pt´ sq´ηF pX1,r

s q ´ F pX2,rs qHγ´η ds

ˇ

ˇ

ˇ

ˇ

2ff

` 2

ż t

0

pt´ sq´η E“

1ΩXtsăρru BpX1,rs q ´BpX

2,rs q

2HSpU,Hγ´η2q

‰

ds.

(7.25)

This shows that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď 2

ˆż t

0

pt´ sq´η ds

˙ż t

0

pt´ sq´η E“

1tsăρru F pX1,rs q ´ F pX

2,rs q

2Hγ´η

‰

ds

` 2

ż t

0

pt´ sq´η E“

1tsăρru BpX1,rs q ´BpX

2,rs q

2HSpU,Hγ´η2q

‰

ds.

(7.26)

Hence, we obtain that for all r P p0,8q, t P r0, T s it holds that

E“

X1,rt ´X2,r

t 2Hγ

‰

ď2T p1´ηq

p1´ ηq

ż t

0

|F |txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,Hγ´ηq

pt´ sqηE“

X1,rs ´X2,r

s 2Hγ

‰

ds

` 2

ż t

0

|B|txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,HSpU,Hγ´η2qq

pt´ sqηE“

X1,rs ´X2,r

s 2Hγ

‰

ds.

(7.27)

Combining this with (7.23) allows us to apply Corollary 1.4.6 to obtain that for allt P r0, T s, r P p0,8q it holds that E

“

X1,rt ´ X2,r

t 2Hγ

‰

“ 0. Monotone convergencehence proves that for all t P r0, T s it holds that

E“

1ttďmintτ1,τ2uuX1t ´X

2t

2Hγ

‰

“ E

”

limrÑ8

1ttďρruX1t ´X

2t

2Hγ

ı

“ limrÑ8

E“

1ttďρruX1t ´X

2t

2Hγ

‰

“ limrÑ8

E“

X1,rt ´X2,r

t 2Hγ

‰

“ 0.(7.28)

This proves that for all t P r0, T s it holds P-a.s. that 1ttďmintτ1,τ2uu rX1t ´ X2

t s “ 0.Combining this with the fact that for every ω P Ω it holds that the function r0, T s Qt ÞÑ 1ttďmintτ1pωq,τ2pωquu rX

1t pωq ´X

2t pωqs P Hγ is left-continuous ensures that

P`

@ t P r0, T s : X1mintt,τ1,τ2u

“ X2mintt,τ1,τ2u

˘

“ P`

@ t P r0, T s : 1ttďmintτ1,τ2uu rX1t ´X

2t s “ 0

˘

“ P`

@ t P r0, T s XQ : 1ttďmintτ1,τ2uu rX1t ´X

2t s “ 0

˘

“ 1.

(7.29)



Corollary 7.1.8 is an immediate consequence from Proposition 7.1.7.

Corollary 7.1.8. Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, let T P p0,8q, γ P R, η P r0, 1q, F P CpHγ, Hγ´ηq, B P CpHγ, HSpU,Hγ´η2qq

satisfy for all bounded sets E Ď Hγ that |F |E|C0,1pE,Hγ´ηq ` |B|E|C0,1pE,HSpU,Hγ´η2qq ă

8, let pΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, and letXk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s-adapted stochastic processes withcontinuous sample paths such that for all k P t1, 2u, t P r0, T s it holds P-a.s. that

Xkt “ etAX1

0 `

ż t

0

ept´sqAF pXks q ds`

ż t

0

ept´sqABpXks q dWs. (7.30)

Then P`

@ t P r0, T s : X1t “ X2

t

˘

“ 1.

7.1.5 Existence and regularity of mild solutions of SPDEs

Theorem 7.1.9. Assume the setting in Subsection 7.1.2. Then there exists an upto modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆ Ω Ñ Hγ

which satisfies suptPr0,T s XtLppP;¨Hγ qă 8 and which is a mild solution of the SPDE


Theorem 7.1.9 can be proved by a standard fixed point argument; see, e.g., Chap-ter 5 in [14].

7.1.6 A priori bounds for mild solutions of SPDEs

Definition 7.1.10. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normed K-vector spaces. Then we denote by ¨LippV,W q : MpV,W q Ñ r0,8s the mapping with

the property that for all f PMpV,W q it holds that

fLippV,W q “ fp0qW ` |f |C0,1pV,W q . (7.32)


Proposition 7.1.11. Assume the setting in Subsection 7.1.2 and let X : r0, T sˆΩ ÑHγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a mild solution of the SPDE


Then

suptPr0,T s

XtLppP;¨Hγ qď?

2 max

1, ξLppP;¨Hγ q

(

¨ Ep1´ηq„

T 1´η?

2 F LippHγ,Hγ´ηq?1´η

`a

T 1´ηppp´ 1q BLippHγ ,HSpU,Hγ´η2qq

ă 8.(7.34)

Proof of Proposition 7.1.11. Observe that Theorem 6.2.32 implies that for all t Pr0, T s it holds that

XtLppP;¨Hγ qď X0LppP;¨Hγ q

`

ż t

0

›

›eApt´sq F pXsq›

›

LppP;¨Hγ qds

`

c

p pp´ 1q

2

„ż t

0

›

›eApt´sqBpXsq›

›

2

LppP;¨HSpU,Hγ qqds

12

ď X0LppP;¨Hγ q`

„

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η F pXsq2LppP;¨Hγ´η

qds

12

`

c

p pp´ 1q

2

„ż t

0

pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q

qds

12

.

(7.35)

This shows that for all t P r0, T s it holds that

XtLppP;¨Hγ qď X0LppP;¨Hγ q

`

„ż t

0

pt´ sq´η max!

1, Xs2LppP;¨Hγ q

)

ds

12

¨

«

F LippHγ ,Hγ´ηq

d

T p1´ηq

p1´ ηq` BLippHγ ,HSpU,Hγ´η2qqq

c

p pp´ 1q

2

ff

.

(7.36)


This proves that for all t P r0, T s it holds that

max!

1, Xt2LppP;¨Hγ q

)

ď 2 max!

1, X02LppP;¨Hγ q

)

`

ż t

0

pt´ sq´η max!

1, Xs2LppP;¨Hγ q

)

ds

¨

«

F LippHγ ,Hγ´ηq

d

2T p1´ηq

p1´ ηq` BLippHγ ,HSpU,Hγ´η2qq

a

p pp´ 1q

ff2

.

(7.37)

An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.11.

Definition 7.1.12. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normed K-vector spaces. Then we denote by ¨LGpV,W q : MpV,W q Ñ r0,8s the mapping with

the property that for all f PMpV,W q it holds that

fLGpV,W q “ supvPV

„

fpvqWmaxt1, vV u

. (7.38)




Then

suptPr0,T s

›

›max

1, XtHγ

(›

›

LppP;|¨|qď?

2›

›max

1, ξHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?

2 F LGpHγ,Hγ´ηq?1´η

`a

T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq

ă 8.(7.40)

Proof of Proposition 7.1.13. Observe that Theorem 6.2.32 implies that for all t P


r0, T s it holds that

›

›max

1, XtHγ

(›

›

LppP;|¨|q

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|q`

ż t

0

›

›eApt´sq F pXsq›

›

LppP;¨Hγ qds

`

c

p pp´ 1q

2

„ż t

0

›

›eApt´sqBpXsq›

›

2


12

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|q`

„

tp1´ηq

p1´ ηq

ż t

0

pt´ sq´η F pXsq2LppP;¨Hγ´η

qds

12

`

c

p pp´ 1q

2

„ż t

0

pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q

qds

12

.

(7.41)

This shows that for all t P r0, T s it holds that

›

›max

1, XtHγ

(›

›

LppP;|¨|q

ď›

›max

1, X0Hγ

(›

›

LppP;|¨|q`

„ż t

0

pt´ sq´η›

›max

1, XsHγ

(›

›

2

LppP;|¨|qds

12

¨

«

F LGpHγ ,Hγ´ηq

d

T p1´ηq

p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qqq

c

p pp´ 1q

2

ff

.

(7.42)


›

›max

1, XtHγ

(›

›

2

LppP;|¨|qď 2

›

›max

1, X0Hγ

(›

›

2

LppP;|¨|q

`

ż t

0

pt´ sq´η›

›max

1, XsHγ

(›

›

2

LppP;|¨|qds

¨

«

F LGpHγ ,Hγ´ηq

d

2T p1´ηq

p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qq

a

p pp´ 1q

ff2

.

(7.43)

An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.13.


Proposition 7.1.14 (Solution process of the SPDE enjoys more regularity than theintial value). Assume the setting in Subsection 7.1.2 and let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q



Then it holds for all t P r0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Xt ´ eAtX0 P

Hr

˘

“ 1 and

›

›Xt ´ eAtX0

›

›

LppP;¨Hr qď max

!

1, supsPr0,T s XsLppP;¨Hγ q

)

¨

«

F LippHγ ,Hγ´ηqtp1`γ´η´rq

p1` γ ´ η ´ rq`

a

p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq

?2 p1` 2β ´ 2rq12

ff

ă 8

(7.45)

and it holds for all t P p0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`

Xt P Hr

˘

“ 1and

XtLppP;¨Hr qď

X0LppP;¨Hγ q

tpr´γq`max

!

1, supsPr0,T s XsLppP;¨Hγ q

)

¨

„

F LippHγ,Hγ´ηqtp1`γ´η´rq

p1`γ´η´rq`

?p pp´1q BLippHγ,HSpU,Hβqq

tp12`β´rq

?2 p1`2β´2rq12

ă 8.(7.46)

Proof of Proposition 7.1.14. First of all, recall that for all t P r0, T s it holds P-a.s.that

Xt ´ eAtX0 “

ż t

0

eApt´sqF pXsq ds`

ż t

0


Moreover, note that Theorem 4.7.6 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηq itholds that

ż t

0

›

›eApt´sqF pXsq›

›

LppP;¨Hr qds

ď F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

„ż t

0

pt´ sqpγ´η´rq ds

“ F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

tp1`γ´η´rq

p1` γ ´ η ´ rqă 8.

(7.48)


Moreover, observe that Theorem 4.7.6 ensures that for all t P r0, T s, r P rγ, β ` 12q

it holds that„ż t

0

›

›eApt´sqBpXsq›

›

2

LppP;¨HSpU,Hrqqds

12

ď BLippHγ ,HSpU,Hβqqmax

#

1, supsPr0,T s

XsLppP;¨Hγ q

+

„ż t

0

pt´ sqp2β´2rq ds

12


#

1, supsPr0,T s

XsLppP;¨Hγ q

+

tp12`β´rq

p1` 2β ´ 2rq12ă 8

(7.49)

Combining (7.47), (7.48), and (7.49) with Theorem 4.7.6 completes the proof ofProposition 7.1.14.

7.1.7 Temporal-regularity of solution processes of SPDEs

Exercise 7.1.15. Assume the setting in Subsection 7.1.2 and let X : r0, T s ˆ Ω Ñ

Hγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q



Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`

0,mint1`γ´η´r, 12`β´ru˘

it holds that

supt1,t2Pr0,T st1‰t2

¨

˝

›

›

`

Xt1 ´ et1AX0

˘

´`

Xt2 ´ et2AX0

˘›

›

LppP;¨Hr q

|t1 ´ t2|ε

˛

‚ă 8. (7.51)

Exercise 7.1.16. Assume the setting in Subsection 7.1.2, let δ P rγ,8q, assumethat ξ P LppP; ¨Hδq, and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications uniquepFtqtPr0,T s-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8

and which is a mild solution of the SPDE


Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`

0,mint1`γ´η´r, 12`β´ru˘

it holds that

supt1,t2Pr0,T st1‰t2

¨

˝

|mintt1, t2u|maxtr`ε´δ,0u

Xt1 ´Xt2LppP;¨Hr q

|t1 ´ t2|ε

˛

‚ă 8. (7.53)


7.1.8 Existence of continuous solutions

See, e.g., Theorem 7.1 in Van Neerven et al. [28] for a more general result.

Proposition 7.1.17. Let pH, x¨, ÿH , ¨Hq and pU, x¨, ÿU , Üq be separable R-Hilbertspaces, let H Ď H be a non-empty orthonormal basis of H, let T P p0,8q, ρ P R, letpΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, let pWtqtPr0,T s be anIdU -cylindrical pFtqtPr0,T s-Wiener process, let A : DpAq Ď H Ñ H be a diagonal linearoperator which satisfies sup

`

t´1u Y σP pA ´ ρq˘

ă 0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R,be a family of interpolation spaces associated to ρ ´ A, and let γ P R, η P r0, 1q,F P LippHγ, Hγ´ηq, B P LippHγ, HSpU,Hγ´η2qq, ξ P MpF0,BpHγqq. Then thereexists an pFtqtPr0,T s-adapted stochastic processes with continuous sample paths which

satisfies that for all t P r0, T s it holds P-a.s. that Xt “ etAξ `şt

0ept´sqAF pXsq ds `

şt

0ept´sqABpXsq dWs and which satisfies

suptPr0,T s

›

›max

1, XtHγ

(›

›

LppP;|¨|Rqď?

2›

›max

1, ξHγ(›

›

LppP;|¨|Rq

¨ Ep1´ηq„

T 1´η?


à


.(7.54)

Proof of Proposition 7.1.17. Throughout this proof let Ωn P F0, n P N0, be thesets with the property that for all n P N0 it holds that Ωn “ tξHγ ă nu and letξn : Ω Ñ Hγ, n P N, be the mappings with the property that for all n P N it holdsthat ξn “ ξ1Ωn . Note that for all q P r0,8q, n P N it holds that E

“

ξnqHγ

‰

ď nq ă 8.Theorem 7.1.9, Exercise 7.1.15, and the Kolmogorov-Chentsov theorem hence ensurethat there exist pFtqtPr0,T s-adapted stochastic processes with continuous sample pathsXn : r0, T s ˆ Ω Ñ Hγ, n P N, which satisfy suptPr0,T sE

“

Xnt Hγ

‰

ă 8 and whichsatisfy that for all n P N, t P r0, T s it holds P-a.s. that

Xnt “ etAξn `

ż t

0

ept´sqAF pXns q ds`

ż t

0

ept´sqABpXns q dWs. (7.55)

Observe that for all k P N, n,m P tk, k ` 1, . . . u, t P r0, T s it holds P-a.s. that

1Ωk rXnt ´X

mt s “

ż t

0

ept´sqA1Ωk

“

F`

1ΩkXns

˘

´ F`

1ΩkXms

˘‰

ds

`

ż t

0

ept´sqA1Ωk

“

B`

1ΩkXns

˘

´B`

1ΩkXms

˘‰

dWs.

(7.56)


We can hence apply Proposition 2.1 in [15] to obtain that for all k P N, n,m P

tk, k ` 1, . . . u it holds that

suptPr0,T s

1Ωk rXnt ´X

mt sLppP;Hγq

“ 0. (7.57)

This implies that

P

˜

@ k P N : @n,m P tk, k ` 1, . . . u : 1Ωk

«

suptPr0,T s

Xnt ´X

mt Hγ

ff

“ 0

¸

“ 1. (7.58)

Next let Y : r0, T s ˆ Ω Ñ Hγ be the mapping with the property that for all pt, ωq Pr0, T s ˆ Ω it holds that

Ytpωq “8ÿ

n“1

Xnt pωq ¨ 1ΩnzΩn´1pωq. (7.59)

Note that for all n P N it holds that

1Ωn suptPr0,T s

Yt ´Xnt Hγ “ sup

tPr0,T s

1ΩnYt ´ 1ΩnXnt Hγ

“ suptPr0,T s

›

›

›

›

›

«

nÿ

k“1

1ΩkzΩk´1Xkt

ff

´ 1ΩnXnt

›

›

›

›

›

Hγ

“ suptPr0,T s

›

›

›

›

›

nÿ

k“1

1ΩkzΩk´1

“

Xkt ´X

nt

‰

›

›

›

›

›

Hγ

“

nÿ

k“1

1ΩkzΩk´1

«

1Ωk suptPr0,T s

›

›Xkt ´X

nt

›

›

Hγ

ff

(7.60)

This and (7.58) show that

P

˜

@n P N : 1Ωn suptPr0,T s

Yt ´Xnt Hγ “ 0

¸

“ 1. (7.61)

Hence, we obtain that for all n P N, t P r0, T s it holds P-a.s. that

1ΩnYt “ 1ΩnXnt

“ 1Ωn

„

etAξn `

ż t

0

ept´sqAF pXns q ds`

ż t

0

ept´sqABpXns q dWs

“ 1Ωn

„

etAξ `

ż t

0

ept´sqA1ΩnF pXns q ds`

ż t

0

ept´sqA1ΩnBpXns q dWs

“ 1Ωn

„

etAξ `

ż t

0

ept´sqAF pYsq ds`

ż t

0

ept´sqABpYsq dWs

.

(7.62)


This implies that for all t P r0, T s it holds P-a.s. that

Yt “ etAξ `

ż t

0

ept´sqAF pYsq ds`

ż t

0

ept´sqABpYsq dWs. (7.63)

Next note that (7.61) and Proposition 7.1.13 ensure that for all n P N it holds that

suptPr0,T s

›

›max

1, 1ΩnYtHγ(›

›

LppP;|¨|q“ sup

tPr0,T s

›

›max

1, 1ΩnXnt Hγ

(›

›

LppP;|¨|q

ď suptPr0,T s

›

›max

1, Xnt Hγ

(›

›

LppP;|¨|qď?

2›

›max

1, ξnHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


à


.

(7.64)

This and Fatou’s lemma imply that for all t P r0, T s it holds that

›

›max

1, YtHγ(›

›

LppP;|¨|q“

›

›

›lim infnÑ8

max

1, 1ΩnYtHγ(

›

›

›

LppP;|¨|q

ď lim infnÑ8

›

›max

1, 1ΩnYtHγ(›

›

LppP;|¨|qď?

2›

›max

1, ξHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


à


.

(7.65)


7.2 Examples of SPDEs

7.2.1 Second order SPDEs

Let T, ϑ P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pU, 〈¨, ¨〉U , Üq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, ξ P

L2pP|F0 ; ¨Hq, let pWtqtPr0,T s be a cylindrical IdU -Wiener process w.r.t. pFtqtPr0,T s,let pekqkPN Ď H satisfy that for all k P N and Borelp0,1q-almost all x P p0, 1q it holdsthat

ekpxq “?

2 sinpkπxq, (7.66)

let A : DpAq Ď H Ñ H be a linear operator with the property that

DpAq “

#

v P H :8ÿ

k“1

k4|〈ek, v〉H |

2Ră 8

+

(7.67)

7.2. EXAMPLES OF SPDES 201


Av “8ÿ

k“1

´ϑπ2k2 〈ek, v〉H ek, (7.68)

let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to Á,let f, b : p0, 1q ˆ RÑ R be Bpp0, 1q ˆRq/BpRq-measurable functions withż 1

0

|fpx, 0q|2 ` |bpx, 0q|2 dx` supxPp0,1q

supy1,y2PRy1‰y2

|fpx,y1q´fpx,y2q|`|bpx,y1q´bpx,y2q||y1ý2|

ă 8, (7.69)

and let F : H Ñ H be the function with the property that for all v P H andBorelp0,1q-almost all x P p0, 1q it holds that

`

F pvq˘

pxq “ fpx, vpxqq. Observe thatthe linear operator A is the Laplace operator with Dirichlet boundary conditions onL2pBorelp0,1q; |¨|Rq multiplied by ϑ. and let β P p´12,´14q. Next observe that for allv P H it holds that

8ÿ

k“1

bp¨, vp¨qq ekp¨q2Hβ“

8ÿ

k“1

›

›pÁqβ`

bp¨, vp¨qq ekp¨q˘›

›

2

H

“

8ÿ

l,k“1

ˇ

ˇ

⟨el, pÁq

β`

bp¨, vp¨qq ekp¨q˘⟩

H

ˇ

ˇ

2

R“

8ÿ

l,k“1

ˇ

ˇ

⟨pÁqβel, bp¨, vp¨qq ekp¨q

⟩H

ˇ

ˇ

2

R

“

8ÿ

l,k“1

›

›pÁqβel›

›

2

H|〈el, bp¨, vp¨qq ekp¨q〉H |

2R

“ÿ

lPN

›

›pÁqβel›

›

2

H

«

ÿ

kPN

|〈ek, bp¨, vp¨qq elp¨q〉H |2

ff

“ÿ

lPN

›

›pÁqβel›

›

2

Hbp¨, vp¨qq elp¨q

2H ď 2 bp¨, vp¨qq2H

«

ÿ

lPN

›

›pÁqβel›

›

2

H

ff

“ 2›

›pÁqβ›

›

2

HSpHqbp¨, vp¨qq2H ă 8.

(7.70)

This and Proposition 2.5.19 ensure that there exists a unique mapping B : H Ñ

HSpH,Hβq which satisfies that for all v P H, u P Cpr0, 1s,Rq and Borelp0,1q-almostall x P p0, 1q it holds that

`

Bpvqu˘

pxq “ bpx, vpxqq ¨ upxq. (7.71)

In addition, (7.70) implies that for all v P H it holds that

BpvqHSpU,Hβq ď?

2›

›pÁqβ›

›

HSpHqbp¨, vp¨qqH ă 8. (7.72)


Moreover, note that for all v, w P H it holds that

Bpvq ´Bpwq2HSpU,Hβq “ÿ

kPN

rBpvq ´Bpwqs ek2Hβ

“ÿ

kPN

›

›pÁqβ rrBpvq ´Bpwqs eks›

›

2

H

“ÿ

k,lPN

ˇ

ˇ

⟨el, pÁq

βrrBpvq ´Bpwqs eks

⟩H

ˇ

ˇ

2

“ÿ

k,lPN

›

›pÁqβel›

›

2

H|〈el, rBpvq ´Bpwqs ek〉H |

2 .

(7.73)

This proves that for all v, w P H it holds that

Bpvq ´Bpwq2HSpU,Hβq “ÿ

k,lPN

›

›pÁqβel›

›

2

H|〈ek, rBpvq ´Bpwqs el〉H |

2

“ÿ

lPN

›

›pÁqβel›

›

2

HrBpvq ´Bpwqs el

2H

ď 2ÿ

lPN

›

›pÁqβel›

›

2

Hbp¨, vp¨qq ´ bp¨, wp¨qq2H

ď 2›

›pÁqβ›

›

2

HSpHq

»

– supxPp0,1q

supy1,y2PRy1‰y2

|bpx, y1q ´ bpx, y2q|

|y1 ´ y2|

fi

fl

2

v ´ w2H ă 8.

(7.74)

This shows that the mapping B : H Ñ HSpH,Hβq in (7.71) is an element of the setC0,1pH,HSpH,Hβqq. We can hence apply Theorem 7.1.9 to obtain that there existsan up to modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆΩ Ñ H which satisfies suptPr0,T sE

“

Xt2H

‰

ă 8 and which is a mild solution of theSPDE


The stochastic process X is thus a mild solution of the SPDE

dXtpxq “”

ϑ B2

Bx2Xtpxq ` fpx,Xtpxqq

ı

dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0

(7.76)for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q. For example, in thecase where b fulfills @x P p0, 1q, y P R : bpx, yq “ 1, the SPDE (7.76) reduces to theSPDE with additive noise

dXtpxq “”

ϑ B2

Bx2Xtpxq ` fpx,Xtpxqq

ı

dt` dWtpxq, Xtp0q “ Xtp1q “ 0 (7.77)

7.2. EXAMPLES OF SPDES 203

for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q and in the case where fand b fulfill @x P p0, 1q, y P R : fpx, yq “ 0 and bpx, yq “ y, the SPDE (7.76) reducesto the stochastic heat equation with linear multiplicative noise

dXtpxq “”

ϑ B2

Bx2Xtpxq

ı

dt`Xtpxq dWtpxq (7.78)

for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q. In the literature theSPDE (7.78) is referred to as continuous version of the parabolic Anderson model.

Question 7.2.1. Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, letpH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let ξ P

H, let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1q ˆRÑ R be a globally Lipschitz continuous function, let X : r0, T s ˆΩ Ñ H be a mildsolution process of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq

for x P p0, 1q, t P r0, T s, let A : DpAq Ď H Ñ H be the Laplace operator withDirichlet boundary conditions on H, and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to ´A.

(i) For which r P R does it holds that @ t P r0, T s : PpXt P Hrq “ 1?

(ii) For which r P R does it holds that @ t P p0, T s : PpXt P Hrq “ 1?

Exercise 7.2.2 (Variances). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq,

let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T sˆΩ Ñ H be a mild solution process of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ 0 (7.79)

for x P p0, 1q, t P r0, T s. Prove or disprove the following statement: It holds thatş1

0E“

|XT pxq|2‰

dx “ř8

n“11´e´2π2n2T

π2n2 .

Part IV

Numerical Analysis of SPDEs

205

Chapter 8

Strong numerical approximationsfor SPDEs

Consider the setting of Section 7.1.2. If one wants to simulate a solution process ofan SPDE on a computer approximatively, then one needs to discretize the possiblyinfinite dimensional Hilbert space H (spatial approximations), the possibly infinitedimensional Hilbert space U (noise approximations) as well as the time intervalr0, T s (temporal approximations). Section 8.1 deals with spatial approximations forSPDEs, Section 8.2 analyses temporal numerical approximations for SPDEs, Sec-tion 8.3 considers noise approximations for SPDEs, and Section 8.4 combines spatial(Section 8.1), temporal (Section 8.2), and noise approximations (Section 8.3) to ob-tain full-discrete numerical approximations for SPDEs.

8.1 Spatial spectral Galerkin approximations for

SPDEs

8.1.1 Galerkin projections

We study Galerkin approximations in Hilbert spaces. For this we recall the conceptof a projection of a Hilbert space on a subspace in Definition 3.3.12 above.

207

208 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES

Lemma 8.1.1 (Representations of projections in Hilbert spaces). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, let U Ď H be a closed subspace of H, letB Ď U be an orthonormal basis of U , let PU P LpHq be the projection of H on U ,and let v P H. Then

PUpvq “ P spanpBq pvq “ÿ

bPB

〈b, v〉H b. (8.1)

8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 209

Example 8.1.2. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, letB Ď H be an orthonormal basis of H, let In Ď B, n P N, be a non-decreasingsequence of subsets of B which satisfies YnPNIn “ B, and let v P H. Then it holdsfor all n P N that

›

›

›

›

›

v ´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“

›

›

›

›

›

«

ÿ

bPB

〈b, v〉H b

ff

´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“

›

›

›

›

›

›

ÿ

bPBzIn

〈b, v〉H b

›

›

›

›

›

›

H

“

d

ÿ

bPBzIn

〈b, v〉H b2H “

d

ÿ

bPBzIn

|〈b, v〉H |2.

(8.2)

The fact thatř

bPB |〈b, v〉H |2“ v2H ă 8 and the assumption that In Ď B, n P N, is

non-decreasing with the property that YnPNIn “ B hence imply that

limnÑ8

›

›

›

›

›

v ´

«

ÿ

bPIn

〈b, v〉H b

ff›

›

›

›

›

H

“ 0. (8.3)

For sufficiently large n P N it thus holds thatÿ

bPIn

〈b, v〉H b (8.4)

is, in the sense of (8.3), a good approximation of v P H. For example, assume thatK “ R, assume that

pH, 〈¨, ¨〉H , ¨Hq “`

L2pBorelp0,1q; |¨|q, ¨L2pBorelp0,1q;|¨|q

, 〈¨, ¨〉L2pBorelp0,1q;|¨|q

˘

, (8.5)

let en P H, n P N, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q it

holds that enpxq “?

2 sinpnπxq, assume that B “ te1, e2, . . . u and assume that forall n P N it holds that In “ te1, e2, . . . , enu. Then

limnÑ8

ż 1

0

ˇ

ˇ

ˇ

ˇ

vpxq ´nř

k“1

2 sinpkπxq

„

1

∫0

sinpkπyq vpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx “ 0 (8.6)

and for sufficiently large n P N it holds that the function

nř

k“1

2 sinpkπxq

„

1

∫0

sinpkπyq vpyq dy

, x P p0, 1q, (8.7)

is, in the sense of (8.6), a good approximation of v P H.


Example 8.1.2 above illustrates how suitable finite-dimensional approximationsof vectors in an infinite dimensional Hilbert space can be obtained; see (8.4) in Ex-ample 8.1.2. Equation (8.3) in Example 8.1.2 also shows that these approximationsin finite dimensional subspaces of the Hilbert space converge to the original vectorin the infinite dimensional Hilbert space. In Proposition 8.1.4 below we intend toprovide more information about how fast the finite dimensional approximations con-verge to the vector in the infinite dimensional Hilbert space. In the formulation ofProposition 8.1.4 the following lemma is used.

Lemma 8.1.3. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with infpσP pAqq ą 0,let B Ď H be an orthonormal basis of H, let λ : B Ñ p0,8q be a function with theproperty that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(8.8)


bPB λb 〈b, v〉H b, letpHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to A, andlet r P R, v P Hr, b P B. Then

〈b, v〉H b “@

bpλbqr

, vD

Hr

bpλbqr

P Hr. (8.9)

The proof of Lemma 8.1.3 is clear and therefore omitted. Instead we now formu-late the main result of this subsection.


Proposition 8.1.4 (A central idea for spectral Galerkin approximations). Let K PtR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, let A : DpAq Ď H Ñ H be a sym-metric diagonal linear operator with infpσP pAqq ą 0, let B Ď H be an orthonormalbasis of H, let λ : BÑ p0,8q be a function with the property that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(8.10)


bPB λb 〈b, v〉H b, letpHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to A, andlet r P R, ρ P r0,8q, I P PpBq, πI P LpHrq satisfy that for all v P Hr it holds that

πIpvq “ÿ

bPI

〈b, v〉H b “ÿ

bPI

@

bpλbqr

, vD

Hr

bpλbqr

“ PspanpIq

Hr pvq P Hr. (8.11)

Then it holds for all v P Hr`ρ that

v ´ πIpvqHr ď›

›A´ρ pIdHr ´πIq›

›

LpHrqvHr`ρ “

„

infbPBzI

λb

´ρ

vHr`ρ . (8.12)

Proof of Proposition 8.1.4. Observe that Proposition 3.5.5 implies that

›

›A´ρ pIdHr ´πIq›

›

LpHrq“

›

›

›

›

›

›

A´ρ

¨

˝

ÿ

bPBzI

@

bpλbqr

, p¨qD

Hr

bpλbqr

˛

‚

›

›

›

›

›

›

LpHrq

“

›

›

›

›

›

›

ÿ

bPBzI

1pλbqρ

@

bpλbqr

, p¨qD

Hr

bpλbqr

›

›

›

›

›

›

LpHrq

“ supbPBzI

”

1pλbqρ

ı

“

„

infbPBzI

λb

´ρ

.

(8.13)


In a number of cases the right hand side of estimate (8.12) converges to zero witha polynomial rate of convergence. This is illustrated in the next example.


Example 8.1.5 (The Laplace operator with Dirichlet boundary conditions). LetA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator withDirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R,be a family of interpolation spaces associated to Á, let r P R, ρ P r0,8q, and leten P L

2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all

x P p0, 1q it holds that enpxq “?

2 sinpnπxq. Then Proposition 8.1.4 proves that forall v P Hr`ρ, n P N it holds that

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

ďvHr`ρπ2ρ n2ρ

ďvHr`ρn2ρ

. (8.14)

Note that, in the setting of Example 8.1.5, it holds for all v P Hr`ρ that

supnPN

˜

n2ρ

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

ă 8. (8.15)

The polynomial convergence rate 2ρ in (8.14) and (8.15) can, in general, not beimproved. This is the subject of the next exercise.

Exercise 8.1.6 (Lower bounds on the convergence speed of spectral Galerkin projec-tions). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace opera-tor with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be a family of interpolation spaces associated to Á, let r P R, ρ P r0,8q, andlet en P L

2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all

x P p0, 1q it holds that enpxq “?

2 sinpnπxq. Give an example of a vector v P Hr`ρ

such that for all ε P p0,8q it holds that

supnPN

˜

np2ρ`εq

›

›

›

›

›

v ńÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

“ 8. (8.16)

The next result, Corollary 8.1.7, specialises Proposition 8.1.4 to the case wherethe vector in the possibly infinite dimensional Hilbert space is the solution processof some SPDE at a fixed time instance.


Corollary 8.1.7 (Galerkin projections for SPDEs). Assume the setting in Subsec-tion 7.1.2, let B Ď H be an orthonormal basis of H, let λ : BÑ p´8, 0q be a functionwith the property that

DpAq “

#

v P H :ÿ

bPB

|λb 〈b, v〉H |2ă 8

+

(8.17)


bPB λb 〈b, v〉H b, letr P p´8, γs, I P PpBq, πI P LpHrq satisfy that for all v P Hr it holds that

πIpvq “ÿ

bPI

〈b, v〉H b “ÿ

bPI

@

b|λb|

r , vD

Hr

b|λb|

r “ PspanpIq

Hr pvq P Hr, (8.18)

and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications unique pFtqtPr0,T s-predictablestochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8 and which is a mildsolution of the SPDE


Then

suptPr0,T s

Xt ´ πIpXtqLppP;¨Hr qď

„

infbPBzI

|λb|

pr´γq

XCpr0,T s,LppP;¨Hγ qqă 8. (8.20)

Corollary 8.1.7 is an immediate consequence from Proposition 8.1.4 and Propo-sition 7.1.11.

Question 8.1.8. Let f : p0, 1q Ñ R be the function with the property that for allx P p0, 1q it holds that fpxq “ x. For which r P R does it hold that

supNPN

˜

N r

ż 1

0

ˇ

ˇ

ˇ

ˇ

fpxq ´Nř

n“1

2 sinpnπxq1ş

0

sinpnπyq fpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx

¸

ă 8? (8.21)

Question 8.1.9. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq and letv P DpAq. For which r P R does it hold that

supNPN

˜

N r

ż 1

0

ˇ

ˇ

ˇ

ˇ

vpxq ´Nř

n“1

2 sinpnπxq1ş

0

sinpnπyq vpyq dy

ˇ

ˇ

ˇ

ˇ

2

dx

¸

ă 8? (8.22)


8.1.2 Setting

Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letλ : BÑ R be a function, assume that

DpAq “

#

v P H :ÿ

bPB

|λb 〈eb, v〉H |2ă 8

+

, (8.23)

assume that for all v P DpAq it holds that Av “ř

bPB λb 〈b, v〉H b, and let pπIqIPPpBq ĎLpHγ´ηq satisfy that for all v P Hγ´η, I P PpBq it holds that

πIpvq “ÿ

bPI

〈b, v〉H b. (8.24)

The above setting allows us to apply Theorem 7.1.9 to obtain that there existup to modifications unique pFtqtPr0,T s-predictable stochastic processes XI : r0, T s ˆΩ Ñ πIpHγq, I P PpBq, with the property that for all I P PpBq it holds thatsuptPr0,T s X

It LppP;¨Hγ q

ă 8 and with the property that for all I P PpBq, t P r0, T sit holds P-a.s. that

XIt “ eAtπIpξq `

ż t

0

eApt´sqπI`

F pXIs q˘

ds`

ż t

0

eApt´sqπI`

BpXIs q dWs

˘

. (8.25)

8.1.3 A strong numerical approximation result for spectralGalerkin approximations of SPDEs

Lemma 8.1.10. Assume the setting in Section 8.1.2 and let I, J P PpBq. Then

›

›XI´XJ

›

›

Cpr0,T s,LppP;¨Hγ qqď?

2›

›πIzJXI` πJzIX

J›

›

Cpr0,T s,LppP;¨Hγ qq(8.26)

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η

`a

ppp´ 1qT 1´η |πIXJB|C0,1pHγ ,HSpU,Hγ´η2qq

ă 8.

Proof of Lemma 8.1.10. Let F P CpHγ, Hγ´ηq and B P CpHγ, HSpU,Hγ´η2qq be thefunctions with the property that for all v P Hγ, u P U it holds that

F pvq “ πIXJ`

F pvq˘

and Bpvqu “ πIXJ`

Bpvqu˘

. (8.27)


The identity πIπJ “ πIXJ then shows that for all t P r0, T s it holds P-a.s. that

„

XIt ´

ż t

0

eApt´sqF pXIs q ds´

ż t

0

eApt´sqBpXIs q dWs

`

„ż t

0

eApt´sqF pXJs q ds`

ż t

0

eApt´sqBpXJs q dWs ´X

Jt

“

„

XIt ´ πJ

ˆ

eAtπIpξq `

ż t

0

eApt´sqπIF pXIs q ds`

ż t

0

eApt´sqπIBpXIs q dWs

˙

`

„

πI

ˆ

eAtπJpξq `

ż t

0

eApt´sqπJF pXJs q ds`

ż t

0

eApt´sqπJBpXJs q dWs

˙

´XJt

““

IdHγ ´πJ‰

XIt `

“

πI ´ IdHγ‰

XJt “ πIzJX

It ´ πJzIX

Jt .

(8.28)

An application of Proposition 7.1.4 hence proves that

suptPr0,T s

›

›XIt ´X

Jt

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a

ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq

¨?

2 suptPr0,T s

›

›πIzJXIt ´ πJzIX

Jt

›

›

LppP;¨Hγ q.

(8.29)

This and the identity

suptPr0,T s

›

›πIzJXIt ´ πJzIX

Jt

›

›

LppP;¨Hγ q“ sup

tPr0,T s

›

›πIzJXIt ` πJzIX

Jt

›

›

LppP;¨Hγ q(8.30)

complete the proof of Lemma 8.1.10.

The next result, Corollary 8.1.11, is an immediate consequence of Lemma 8.1.10above.

Corollary 8.1.11. Assume the setting in Section 8.1.2, let r P pγ,mintγ`1´η, β`12uq, I P PpBq, and assume that ξ P LppP; ¨Hrq. Then

suptPr0,T s

›

›XB

t ´XIt

›

›

LppP;¨Hγ qď?

2“

infbPBzI |λb|‰pγ´rq

suptPr0,T s

›

›XB

t

›

›

LppP;¨Hr q(8.31)

¨ Ep1´ηq„

T 1´η?


`a

ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq

ă 8.


The next result, Corollary 8.1.12, establishes a certain uniform moment boundfor the processes XI , I P PpBq. Corollary 8.1.12 is an immediate consequence ofProposition 7.1.11 and from Theorem 7.1.9.

Corollary 8.1.12. Assume the setting in Section 8.1.2, let r P rγ,mintγ`1´η, β`12uq, ρ P

“

maxtr ` η ´ γ, 2pr ´ βqu, 1˘

, and assume that ξ P LppP; ¨Hrq. Then

supIPPpBq

›

›XI›

›

Cpr0,T s,LppP;¨Hr qqď?

2 max

1, ξLppP;¨Hr q

(

¨ Ep1´ρq„

T 1´ρ?

2 F LippHr,Hr´ρq?1´ρ

à

T 1´ρppp´ 1q BLippHr,HSpU,Hr´ρ2qq

ă 8.(8.32)

Lemma 8.1.13. Assume the setting in Section 8.1.2, let r P rγ,mintγ`1´η, β`12uq,I, J P PpBq, and assume that ξ P LppP; ¨Hrq. Then

›

›XI´XJ

›

›

Cpr0,T s,LppP;¨Hγ qqď 2

›

›pÁqpγ´rqπI4J›

›

LpHqmaxKPtI,Ju

›

›XK›

›

Cpr0,T s,LppP;¨Hr qq

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η

à


ă 8.


›

›πIzJXI` πJzIX

J›

›

2

Cpr0,T s,LppP;¨Hγ qq“ sup

tPr0,T s

›


Jt

›

›

2

LppP;¨Hγ q

“ suptPr0,T s

›

›pÁqpγ´rqπI4J pÁqpr´γq

“

πIzJXIt ` πJzIX

Jt

‰›

›

2

LppP;¨Hγ q

ď›


›

2

LpHqsuptPr0,T s

›


Jt

›

›

2

LppP;¨Hr q

“›


›

2

LpHqsuptPr0,T s

›

›

›

›


Jt

›

›

2

Hr

›

›

›

Lp2pP;|¨|q

“›


›

2

LpHqsuptPr0,T s

›

›

›

›

›πIzJXIt

›

›

2

Hr`›

›πJzIXJt

›

›

2

Hr

›

›

›

Lp2pP;|¨|q

ď›


›

2

LpHqsuptPr0,T s

„

›

›

›

›

›πIzJXIt

›

›

2

Hr

›

›

›

Lp2pP;|¨|q`

›

›

›

›

›πJzIXJt

›

›

2

Hr

›

›

›

Lp2pP;|¨|q

“›


›

2

LpHqsuptPr0,T s

”

›

›πIzJXIt

›

›

2

LppP;¨Hr q`›

›πJzIXJt

›

›

2

LppP;¨Hr q

ı

.

(8.33)

Combining this with Lemma 8.1.10 completes the proof of Lemma 8.1.13.


Let us illustrate a bit different perspective on Lemma 8.1.13. For this assume thesetting in Section 8.1.2, let dr : PpBq ˆ PpBq Ñ r0,8q, r P p0,8q, be the mappingswith the property that for all I, J P PpBq, r P p0,8q it holds that

drpI, Jq “›

›pÁq´rπI4J›

›

LpHq, (8.34)

let r P pγ,mint1`γ´η, β` 12uq, and let ξ P LppP; ¨Hrq. Exercise 8.1.14 then showsthat the pair pPpBq, drq is a metric space and Lemma 8.1.13, in particular, ensuresthat the mapping1

pPpBq, dr´γq Q I ÞÑ XIP

´

Cpr0, T s, LppP; ¨Hγ qq, ¨Cpr0,T s,LppP;¨Hγ qq

¯

(8.35)

is globally Lipschitz continuous with a Lipschitz constant which is smaller or equalthan

2 ¨

«

supIPPpBq

suptPr0,T s

›

›XI›

›

LppP;¨Hr q

ff

(8.36)

¨ Ep1´ηq„

T 1´η?

2 |πIXJF |CC1pHγ,Hγ´ηq?1´η

à


ă 8.

Exercise 8.1.14. Assume the setting in Section 8.1.2 and let dr : PpBq ˆ PpBq Ñr0,8q, r P p0,8q, be the mappings with the property that for all I, J P PpBq, r Pp0,8q it holds that

drpI, Jq “›

›pÁq´rπI4J›

›

LpHq. (8.37)

Prove that for every r P p0,8q it holds that the pair pPpBq, drq is a metric space.

1Clearly, the domain (the set of arguments) of the mapping (8.35) is not pPpBq, dr´γq but theset PpBq. The notation (8.35) is nonetheless used in order to emphasize the specific metric definedon the set PpBq of arguments. The same comment applies to the co-domain of the mapping (8.35).


Exercise 8.1.15. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq (see Def-inition 3.5.7), let T “ 1, let en P L2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all

n P N and Borelp0,1q-almost all x P p0, 1q it holds that enpxq “?

2 sinpnπxq, letpΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be a cylindrical IdH-Wienerprocess w.r.t. pFtqtPr0,T s, let X : r0, T sˆΩ Ñ H be an pFtqtPr0,T s-predictable stochasticprocess which fulfills that for all t P r0, T s it holds P-a.s. that

Xt “

ż t

0

eApt´sq dWs, (8.38)

let πN P LpHq, N P N, satisfy that for all N P N, v P H it holds that πNpvq “řNn“1 〈en, v〉H en, and let XN : r0, T s ˆΩ Ñ PNpHq, N P N, be pFtqtPr0,T s-predictable

stochastic processes which satisfy that for all N P N, t P r0, T s it holds P-a.s. that

XNt “

ż t

0

πN eApt´sq dWs. (8.39)

Write a Matlab function which plots Monte Carlo approximations of the real num-

bers`

E“

X223

T ´ XNT

2H

‰˘12for N P t20, 21, 22, 23, . . . , 218, 219u. Hint: Use the fact

that for every N P N you can simulate exactly from XNT pPq.

Question 8.1.16 (Convergence speed of spectral Galerkin approximations). Assumethe setting in Section 8.1.2 and assume that ξ P LppP; ¨Hγ`1

q. For which r P R does

it holds that there exists a real number C P R such that for all I P PpBq it holds that

suptPr0,T s

›

›XB

t ´XIt

›

›

LppP;¨Hγ qď C

“

infbPBzI |λb|‰´r

. (8.40)

8.2 Temporal numerical approximations for SPDEs

In this section we present and analyze a few temporal numerical approximations forSPDEs. For this the following notation is used.

Definition 8.2.1 (Round down to the grid). Let t¨uh : R Ñ R, h P p0,8q, be themappings with the property that for all h P p0,8q, t P R it holds that

ttuh “ max`

p´8, ts X t0, h,´h, 2h,´2h, . . . u˘

. (8.41)

8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 219

Definition 8.2.2 (Down to the grid). Let z¨h : RÑ R, h P p0,8q, be the mappingswith the property that for all h P p0,8q, t P R it holds that

zth “ max`

p´8, tq X t0, h,´h, 2h,´2h, . . . u˘

. (8.42)

Using Definition 8.2.1 we will now present in Subsections 8.2.1 and 8.2.3 be-low a few temporal numerical approximation methods for SPDEs and then analyzethe strong approximation errors of one of these approximation methods in Subsec-tion 8.2.4 below.

8.2.1 Euler type approximations for SPDEs

8.2.1.1 Exponential Euler method

Definition 8.2.3 (Exponential Euler approximations). Assume the setting in Sec-tion 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfills that for all n Pt0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ eATN

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

. (8.43)

Then we call Y an exponential Euler approximation for the SPDE


with time step size TN.

Definition 8.2.4 (Naturally-interpolated exponential Euler approximations). As-sume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ be anpFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and which fulfills that forall t P p0, T s it holds P-a.s. that

Yt “ eApt´zthq

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

. (8.45)

Then we call Y a naturally-interpolated exponential Euler approximation for theSPDE


with time step size h.


Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ

be a naturally-interpolated exponential Euler approximation for the SPDE


with time step size h. Then note that for all t P r0, T s it holds P-a.s. that

Yt “ eAtξ `

ż t

0

eApt´tsuhqF`

Ytsuh

˘

ds`

ż t

0

eApt´tsuhqB`

Ytsuh

˘

dWs. (8.48)

Exercise 8.2.5. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P

p0,8q, N P N, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be a cylin-drical IdH-Wiener process w.r.t. pFtqtPr0,T s, let Y : r0, T s ˆ Ω Ñ H be a naturally-interpolated exponential Euler approximation for the SPDE

dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (8.49)

with time step size TN, and let X : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-predictablestochastic process which fulfills that for all t P r0, T s it holds P-a.s. that

Xt “

ż t

0

eApt´sq dWs. (8.50)

Prove that for all r P r0, 14q it holds that`

E“

XT ´ YT 2H

‰˘12ď T r

p1´4rqNr .

8.2.1.2 Accelerated exponential Euler method

Definition 8.2.6 (Accelerated exponential Euler approximations). Assume the set-ting in Section 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ eATN Yn ` A

´1´

eATN ´ IdH

¯

F`

Yn˘

`

ż pn`1qTN

nTN

eAppn`1qTN´sqB

`

Yn˘

dWs.

(8.51)

Then we call Y an accelerated exponential Euler approximation for the SPDE




Definition 8.2.7 (Naturally-interpolated accelerated exponential Euler approxima-tions). Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ

be an pFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and which fulfills thatfor all t P p0, T s it holds P-a.s. that

Yt “ eApt´zthq Yzth `

ż t

zth

eApt´sqF`

Yzth

˘

ds`

ż t

zth

eApt´sqB`

Yzth

˘

dWs. (8.53)

Then we call Y a naturally-interpolated accelerated exponential Euler approximationfor the SPDE




be a naturally-interpolated exponential Euler approximation for the SPDE



Yt “ eAtξ `

ż t

0

eApt´sqF`

Ytsuh

˘

ds`

ż t

0

eApt´sqB`

Ytsuh

˘

dWs. (8.56)

8.2.1.3 Linear-implicit Euler method

Definition 8.2.8 (Linear-implicit Euler approximations). Assume the setting in Sec-tion 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfills that for alln P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´TNA˘´1

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

. (8.57)

Then we call Y a linear-implicit Euler approximation for the SPDE




Definition 8.2.9 (Naturally-interpolated linear-implicit Euler approximations). As-sume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ be anpFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and which fulfills that forall t P p0, T s it holds P-a.s. that

Yt “ pIdH ´pt´ zthqAq´1

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

. (8.59)

Then we call Y a naturally-interpolated linear-implicit Euler approximation for theSPDE




be a naturally-interpolated linear-implicit Euler approximation for the SPDE



Yt “`

IdH ´pt´ ttuhqA˘´1 `

IdH ´hA˘´ttuhh

ξ

`

ż t

0

`


IdH ´hA˘ptsuh´ttuhqh

F`

Ytsuh

˘

ds

`

ż t

0

`



B`

Ytsuh

˘

dWs.

(8.62)

8.2.1.4 Linear-implicit Crank-Nicolson-Euler method

Definition 8.2.10 (Linear-implicit Crank-Nicolson-Euler approximations). Assumethe setting in Section 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´T

2NA˘´1

˜

`

IdH `T

2NA˘

Yn ` F pYnqTN`

ż pn`1qTN

nTN

BpYnq dWs

¸

.

(8.63)

Then we call Y a linear-implicit Crank-Nicolson-Euler approximation for the SPDE




Definition 8.2.11 (Naturally-interpolated linear-implicit Crank-Nicolson-Euler ap-proximations). Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that

Yt “´

IdH ´pt´zthq

2A¯´1

ˆ

´

IdH `pt´zthq

2A¯

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

.

(8.65)

Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Euler approx-imation for the SPDE




be a naturally-interpolated linear-implicit Euler approximation for the SPDE



Yt “`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘´ttuhh

ξ

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘ptsuh´ttuhqh

“

12AYtsuh ` F

`

Ytsuh

˘‰

ds

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


B`

Ytsuh

˘

dWs.

(8.68)

8.2.2 Nonlinearity-stopped Euler type approximations forSPDEs

The next result is a special case of Theorem 2.1 in Hutzenthaler et al. [12] (see also[13]).


Theorem 8.2.12 (Strong and weak divergence of the Euler method for SDEs withsuperlinearly growing coefficients). Let T, ε, p P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let W : r0, T s ˆ Ω Ñ R be a standard Brownian motion w.r.t.pFtqtPr0,T s, let µ, σ PMpBpRq,BpRqq, ξ PMpF0,BpRqq satisfy P

`

σpξq ‰ 0˘

ą 0, letY N : t0, 1, . . . , Nu ˆΩ Ñ R, N P N, satisfy that for all N P N, n P t0, 1, . . . , N ´ 1uit holds that Y N

0 “ ξ and

Y Nn`1 “ Y N

n ` µpY Nn q

TN` σpY N

n q`

Wpn`1qTN ´WnTN

˘

, (8.69)

and assume that for all x P p´8, 1εsYr1ε,8q it holds that |µpxq|`|σpxq| ě ε |x|p1`εq.Then limNÑ8 E

“

|Y NN |

p‰

“ 8.

8.2.2.1 Nonlinearity-stopped exponential Euler method

Definition 8.2.13 (Nonlinearity-stopped exponential Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s.that

Yn`1 “ 1tF pYnq2HαąNTuYn

` 1tF pYnq2HαďNTueA

TN

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

.(8.70)

Then we call Y a nonlinearity-stopped exponential Euler approximation for the SPDE




Definition 8.2.14 (Naturally-interpolated nonlinearity-stopped exponential Eulerapproximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ ´ η, γs,assume that F pHγq Ď Hα, and let Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adaptedstochastic process which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s it holdsP-a.s. that

Yt “ 1tF pYzthq2Hα

ą1hu Yzth (8.72)

` 1tF pYzthq2αď1hu e

Apt´zthq

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

.

Then we call Y a naturally-interpolated nonlinearity-stopped exponential Euler ap-proximation for the SPDE



8.2.2.2 Nonlinearity-stopped linear-implicit Euler method

Definition 8.2.15 (Nonlinearity-stopped linear-implicit Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s.that

Yn`1 “ 1tF pYnq2HαąNTuYn (8.74)

` 1tF pYnq2HαďNTu

`

IdH ´TNA˘´1

˜

Yn ` F`

Yn˘

TN`

ż pn`1qTN

nTN

B`

Yn˘

dWs

¸

.

Then we call Y a nonlinearity-stopped linear-implicit Euler approximation for theSPDE




Definition 8.2.16 (Naturally-interpolated nonlinearity-stopped linear-implicit Eu-ler approximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ´η, γs,assume that F pHγq Ď Hα, and let Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adaptedstochastic processes which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s itholds P-a.s. that

Yt “ 1tF pYzthq2Hα

ą1hu Yzth ` 1tF pYzthq2Hα

ď1hu pIdH ´pt´ zthqAq´1

ˆ

Yzth

` F`

Yzth

˘

pt´ zthq `

ż t

zth

B`

Yzth

˘

dWs

˙

. (8.76)

Then we call Y a naturally-interpolated nonlinearity-stopped linear-implicit Eulerapproximation for the SPDE



8.2.3 Milstein type approximations for SPDEs

8.2.3.1 Exponential Milstein method

Definition 8.2.17 (Exponential Milstein approximations). Assume the setting inSection 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is continu-ously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ eATN

˜

Yn ` F`

Yn˘

TN`

żpn`1qTN

nTN

B`

Yn˘

dWs

`

żpn`1qTN

nTN

B1`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

.

(8.78)

Then we call Y an exponential Milstein approximation for the SPDE




Definition 8.2.18 (Naturally-interpolated exponential Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, and let Y : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that

Yt “ eApt´zthq

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.80)

Then we call Y a naturally-interpolated exponential Milstein approximation for theSPDE




be a naturally-interpolated exponential Milstein approximation for the SPDE



Yt “ eAtξ `

ż t

0

eApt´tsuhqF`

Ytsuh

˘

ds

`

ż t

0

eApt´tsuhq”

B`

Ytsuh

˘

`B1`

Ytsuh

˘s

∫tsuh

BpYtsuhq dWu

ı

dWs.

(8.83)


8.2.3.2 Linear-implicit Milstein method

Definition 8.2.19 (Linear-implicit Milstein approximations). Assume the settingin Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is contin-uously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´TNA˘´1

˜

Yn ` F`

Yn˘

TN`

żpn`1qTN

nTN

B`

Yn˘

dWs

`

żpn`1qTN

nTN

B1`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

.

(8.84)

Then we call Y a linear-implicit Milstein approximation for the SPDE



Definition 8.2.20 (Naturally-interpolated linear-implicit Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, and let Y : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that

Yt “pIdH ´pt´ zthqAq´1

ˆ

Yzth ` F`

Yzth

˘

pt´ zthq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.86)

Then we call Y a naturally-interpolated linear-implicit Milstein approximation forthe SPDE




be a naturally-interpolated linear-implicit Milstein approximation for the SPDE




Yt “`


IdH ´hA˘´ttuhh

ξ

`

ż t

0

`



F`

Ytsuh

˘

ds

`

ż t

0

`



B`

Ytsuh

˘

dWs

`

ż t

0

`



”

B1`

Yttuh

˘s

∫ttuh

B`

Yttuh

˘

dWu

ı

dWs.

(8.89)

8.2.3.3 Linear-implicit Crank-Nicolson-Milstein method

Definition 8.2.21 (Linear-implicit Crank-Nicolson-Milstein approximations). As-sume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ

HSpU,Hγq is continuously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ andwhich fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “`

IdH ´T

2NA˘´1

˜

`

IdH `T

2NA˘

Yn ` F`

Yn˘

TN`

żpn`1qTN

nTN

B`

Yn˘

dWs

`

żpn`1qTN

nTN

B1`

Yn˘

ˆż s

nTN

B`

Yn˘

dWu

˙

dWs

¸

. (8.90)

Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Milstein ap-proximation for the SPDE




Definition 8.2.22 (Naturally-interpolated linear-implicit Crank-Nicolson-Milsteinapproximations). Assume the setting in Section 7.1.2, assume that γ “ β, assumethat B : Hγ Ñ HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, andlet Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfillsY0 “ ξ and which fulfills that for all t P p0, T s it holds P-a.s. that

Yt “`

IdH ´12pt´ zthqA

˘´1

ˆ

Yzth `12AYzth pt´ zthq ` F

`

Yzth

˘

pt´ zthq

`

ż t

zth

”

B`

Yzth

˘

`B1`

Yzth

˘s

∫zth

B`

Yzth

˘

dWu

ı

dWs

˙

.

(8.92)

Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Milstein ap-proximation for the SPDE



Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ bea naturally-interpolated linear-implicit Crank-Nicolson-Milstein approximation forthe SPDE



Yt “`

IdH ´pt´ttuhq

2A˘´1 `

IdH ´h2A˘´ttuhh

ξ

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


”

12AYtsuh ` F

`

Ytsuh

˘

ı

ds

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


B`

Ytsuh

˘

dWs

`

ż t

0

`

IdH ´pt´ttuhq

2A˘´1 `


”

B1`

Yttuh

˘s

∫ttuh

B`

Yttuh

˘

dWu

ı

dWs.

(8.95)

8.2.4 Strong convergence analysis for exponential Euler ap-proximations

In this subsection we establish strong convergence with suitable rates of convergenceof exponential Euler approximations; see Definition 8.2.3. In this subsection wemainly follow the analysis in Kurniawan [21].


Lemma 8.2.23 (Regularity for the numerical approximations). Assume the settingin Subsection 7.1.2, let N P N, and let Y : r0, T sˆΩ Ñ Hγ be a naturally-interpolatedexponential Euler approximation for the SPDE


with time step size TN. Then

suptPr0,T s

YtLppP;¨Hγ qď max

1,?

2 ξLppP;¨Hγ q

(

(8.97)

¨ E1´η

„

T 1´η?

2 F LippHγ,Hγ´ηq?1´η

`a

T 1´η p pp´ 1q BLippHγ ,HSpU,Hγ´η2qq

ă 8.

Proof of Lemma 8.2.23. Theorem 4.7.6 and Holder’s inequality imply that for allt P r0, T s it holds that

YtLppP;¨Hγ qď ξLppP;¨Hγ q

` F LippHγ ,Hγ´ηq

„

tp1´ηq

p1´ηq

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

` BLippHγ ,HSpU,Hγ´η2qq

„

ppp´1q2

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

.

(8.98)


YtLppP;¨Hγ qď ξLppP;¨Hγ q

`

„ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

12

¨

„

F LippHγ ,Hγ´ηqTp1´ηq2?

1´η` BLippHγ ,HSpU,Hγ´η2qq

?ppp´1q?

2

.

(8.99)

Induction hence proves that for all t P r0, T s it holds that YtLppP;¨Hγ qă 8.

Moreover, combining (8.99) with the estimate that for all a, b P R it holds thatpa` bq2 ď 2a2 ` 2b2 shows that for all t P r0, T s it holds that

max

1, Yt2LppP;¨Hγ q

(

ď max

1, 2 ξ2LppP;¨Hγ q

(

`

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

¨

”

F LippHγ ,Hγ´ηq

?2T

p1´ηq2?


a

p pp´ 1qı2

.

(8.100)


Next note that for all u P r0, T s it holds that

suptPr0,us

ż t

0

pt´ sq´η max

1,›

›YtsuT N

›

›

2

LppP;¨Hγ q

(

ds

ď suptPr0,us

ż t

0

pt´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds

“ suptPr0,us

ż u

u´t

pt´ rs´ pu´ tqsq´η”

supvPr0,s´pu´tqs max

1, Yv2LppP;¨Hγ q

(

ı

ds

“ suptPr0,us

ż u

u´t

pu´ sq´η”

supvPr0,s`t´us max

1, Yv2LppP;¨Hγ q

(

ı

ds

ď suptPr0,us

ż u

u´t

pu´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds

“

ż u

0

pu´ sq´η”

supvPr0,ss max

1, Yv2LppP;¨Hγ q

(

ı

ds.

(8.101)

Putting this into (8.100) proves that for all u P r0, T s it holds that

suptPr0,us max

1, Yt2LppP;¨Hγ q

(

ď max

1, 2 ξ2LppP;¨Hγ q

(

`

ż u

0

pu´ sq´η”

suptPr0,ss max

1, Yt2LppP;¨Hγ q

(

ı

ds

¨

”

F LippHγ ,Hγ´ηq

?2T

p1´ηq2?


a

p pp´ 1qı2

.

(8.102)

Combining this with Corollary 1.4.6 completes the proof of Lemma 8.2.23.


Lemma 8.2.24 (More regularity for the numerical approximations). Assume thesetting in Subsection 7.1.2, let N P N, and let Y : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Euler approximation for the SPDE


with time step size TN. Then it holds for all t P r0, T s, r P rγ,mint1`γ´η, 12`βuqthat P

`

Yt ´ eAtY0 P Hr

˘

“ 1 and

›

›Yt ´ eAtY0

›

›

LppP;¨Hr qď max

#

1, supnPt0,1,...,Nu

YnTNLppP;¨Hγ q

+

¨

«

F LippHγ ,Hγ´ηqtp1`γ´η´rq

p1` γ ´ η ´ rq`

a

p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq

p2` 4β ´ 4rq12

ff

ă 8

(8.104)

and it holds for all t P p0, T s, r P rγ,mint1` γ´ η, 12`βuq that P`

Yt P Hr

˘

“ 1 and

YtLppP;¨Hr qď

X0LppP;¨Hγ q

tpr´γq`max

!

1, supnPt0,1,...,Nu YnTNLppP;¨Hγ q

)

¨

„

F LippHγ,Hγ´ηqtp1`γ´η´rq

p1`γ´η´rq`

?p pp´1q BLippHγ,HSpU,Hβqq

tp12`β´rq

p2`4β´4rq12

ă 8.(8.105)

Proof of Lemma 8.2.24. First of all, recall that for all t P r0, T s it holds P-a.s. that

Yt ´ eAtY0 “

ż t

0

eApt´tsuT N qF pYtsuT Nq ds`

ż t

0

eApt´tsuT N qBpYtsuT Nq dWs. (8.106)

Moreover, note that Theorem 4.7.6 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηq itholds that

ż t

0

›

›

›eApt´tsuT N qF pYtsuT N

q

›

›

›

LppP;¨Hr qds

ď F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

„ż t

0

pt´ sqpγ´η´rq ds

“ F LippHγ ,Hγ´ηqmax

#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

tp1`γ´η´rq

p1` γ ´ η ´ rqă 8.

(8.107)


Furthermore, observe that Theorem 4.7.6 ensures that for all t P r0, T s, r P rγ, β`12q

it holds that

„ż t

0

›

›

›eApt´tsuT N qBpYtsuT N

q

›

›

›

2

LppP;¨HSpU,Hrqqds

12


#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

„ż t

0

pt´ sqp2β´2rq ds

12


#

1, supsPr0,T s

YtsuT NLppP;¨Hγ q

+

tp12`β´rq

p1` 2β ´ 2rq12ă 8.

(8.108)

Combining (8.106), (8.107) and (8.108) with Theorem 4.7.6 and Theorem 6.2.32completes the proof of Lemma 8.2.24.

Lemma 8.2.25 (Regularity of a time integral associated to the numerical approx-imations). Assume the setting in Section 7.1.2, let ρ P r0, 1q, q P r1,8q, and forevery N P N let Y N : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Eulerapproximation for the SPDE


with time step size TN. Then it holds for all r P p´8,mintp1´ρqq, 1´ η, 12` β ´ γuqthat

supNPN

suptPr0,T s

„

N rt

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ă 8. (8.110)

Proof of Lemma 8.2.25. First of all, observe that for all N P N it holds that

suptPr0,TNs

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 suptPp0,T s›

›Y Nt

›

›

LppP;¨Hγ qT p1´ρqq

p1´ ρq1qN p1´ρqq.

(8.111)

Next note that for all r P rγ,mint1 ` γ ´ η, 12 ` βuq, N P N, t P rTN, T s it holds


that

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď 2 supsPp0,T s

›

›Y Ns

›

›

LppP;¨Hγ q

„

TN

∫0pt´ sq´ρ ds

1q

`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff«

t

∫TN

pt´ sq´ρ

`

tsuT N˘q pr´γq

ds

ff1q

ď 2 supsPp0,T s

›

›Y Ns

›

›

LppP;¨Hγ q

T p1´ρqq

N p1´ρqq p1´ ρq1q

`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

„

t´TN

∫0

pt´ TN ´ sq´ρ

sq pr´γqds

1q

.

(8.112)

This implies that that for all r P rγ,mint1` γ´ η, 12`βuq, N P N, t P rTN, T s withρ` qpr ´ γq ď 1 it holds that

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 supsPp0,T s›

›Y Ns

›

›



`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

¨ rt´ TNsr1´ρq´pr´γqs

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

.

(8.113)

Combining this with (8.111) proves that for all r P rγ,mintγ`p1´ρqq, γ`1´η, 12`βuq,


N P N it holds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

2 supsPp0,T s›

›Y Ns

›

›



`

«

supsPrTN,T s

`

tsuT N˘pr´γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ff

¨ T rγ`p1´ρqq´rs

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

.

(8.114)

In the next step we observe that Theorem 6.2.32, Lemma 4.7.7 and Lemma 8.2.24ensure that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holds that

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qď

›

›

›

`

eAps´tsuT N q ´ IdH˘

Y NtsuT N

›

›

›

LppP;¨Hγ q

`

ż s

tsuT N

›

›eAps´tuuT N qF pY NtuuT N

q›

›

LppP;¨Hγ qdu

`

«

p pp´ 1q

2

ż s

tsuT N

›

›

›eAps´tuuT N qBpY N

tuuT Nq

›

›

›

2

LppP;¨HSpU,Hγ qqdu

ff12

ď`

s´ tsuT N˘pr´γq

›

›

›Y N

tsuT N

›

›

›

LppP;¨Hr q

`

ż s

tsuT N

ps´ uq´η›

›

›F pY N

tuuT Nq

›

›

›

LppP;¨Hγ´ηqdu

`

«

p pp´ 1q

2

ż s

tsuT N

ps´ uqp2β´2γq›

›

›BpY N

tuuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqdu

ff12

ă 8.

(8.115)

This implies that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holdsthat›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qď

„

T

N

pr´γqˇ

ˇtsuT Nˇ

ˇ

pγ´rq

«

supuPp0,T s

upr´γq›

›Y Nu

›

›

LppP;¨Hr q

ff

`“

TN

‰p1´ηq F LippHγ,Hγ´ηq

p1´ηqmax

"

1, supnPt1,2,...,Nu›

›Y NnTN

›

›

LppP;¨Hγ q

*

`“

TN

‰p12`β´γq BLippHγ,HSpU,Hβqq

p1`2β´2γq12

”

p pp´1q2

ı12

max

"

1, supnPt1,2,...,Nu›

›Y NnTN

›

›

LppP;¨Hγ q

*

.

(8.116)


This shows that for all N P N, r P rγ,mint1` γ ´ η, 12` βuq it holds that

supsPrTN,T s

„

ˇ

ˇtsuT Nˇ

ˇ

pr´γq›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ q

ď

”

“

TN

‰pr´γq`“

TN

‰p1´ηq Tpr´γq F LippHγ,Hγ´ηq

p1´ηq

`“

TN

‰p12`β´γq Tpr´γq BLippHγ,HSpU,Hβqq

p12`β´γq12

”

p pp´1q4

ı12 ı

¨max

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.117)

Putting this into (8.114) proves that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´η, 12` βuq it holds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

«

2T p1´ρqq


`

”

1`T p1´ηq F LippHγ,Hγ´ηq

p1´ηq`

T p12`β´γq BLippHγ,HSpU,Hβqq

p12`β´γq12

”

p pp´1q4

ı12 ı

¨T p1´ρqq

N pr´γq

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

ff

¨max

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.118)

Hence, we obtain that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´ η, 12 ` βuq itholds that

suptPr0,T s

„

t

∫0pt´ sq´ρ

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

q

LppP;¨Hγ qds

1q

ď

„

52` F LippHγ ,Hγ´ηq ` BLippHγ ,HSpU,Hβqq

?p pp´1q

2

¨

“

B`

1´ ρ, 1´ qpr ´ γq˘‰1q

maxpT 2, 1q

mint1´ ρ, 1´ η, 12` β ´ γuN pr´γqmax

#

1, supvPrγ,rs

supuPp0,T s

upv´γq›

›Y Nu

›

›

LppP;¨Hv q

+

.

(8.119)

This and Lemma 8.2.24 complete the proof of Lemma 8.2.25.

In the next result, Corollary 8.2.26, an estimate for the strong approximationerror of exponential Euler approximations is presented.


Corollary 8.2.26. Assume the setting in Section 7.1.2, let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q


dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (8.120)

let N P N, and let Y : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Eulerapproximation for the SPDE (8.120) with time step size TN. Then

suptPr0,T s

Xt ´ YtLppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a


¨?

2 suptPr0,T s

«

ż t

0

›

›

›eApt´sq

”

F pYsq ´ eAps´tsuT N qF pYtsuT N

q

ı›

›

›

LppP;¨Hγ qds

`

„

p pp´1q2

t

∫0

›

›

›eApt´sq

”

BpYsq ´ eAps´tsuT N qBpYtsuT N

q

ı›

›

›

2


12ff

ă 8.

(8.121)

Corollary 8.2.26 is an immediate consequence of Proposition 7.1.4 and of Theo-rem 6.2.32.

Theorem 8.2.27. Assume the setting in Section 7.1.2, let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q



and for every N P N let Y N : r0, T sˆΩ Ñ Hγ be a naturally-interpolated exponentialEuler approximation for the SPDE (8.122) with time step size TN. Then it holds forall r P

`

´8,mint1´ η, 12` β ´ γu˘

that

supNPN

suptPr0,T s

„

N r›

›Xt ´ YNt

›

›

LppP;¨Hγ q

ă 8. (8.123)


Proof of Theorem 8.2.27. Observe that Theorem 4.7.6 and Lemma 4.7.7 imply thatfor all t P r0, T s, ε P p0, 1´ ηq it holds that

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ e

Aps´tsuT N qF pY NtsuT N

q

ı›

›

›

LppP;¨Hγ qds

ď

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ F pY

NtsuT N

q

ı›

›

›

LppP;¨Hγ qds

`

ż t

0

›

›

›eApt´sq

`

IdH ´eAps´tsuT N q

˘

F pY NtsuT N

q

›

›

›

LppP;¨Hγ qds

ď

ż t

0

pt´ sq´η›

›

›F pY N

s q ´ F pYN

tsuT Nq

›

›

›

LppP;¨Hγ´η qds

`

ż t

0

pt´ sq´η´ε`

s´ tsuT N˘ε›

›

›F pY N

tsuT Nq

›

›

›

LppP;¨Hγ´η qds.

(8.124)

This ensures that for all t P r0, T s, ε P p0, 1´ ηq it holds that

ż t

0

›

›

›eApt´sq

”

F pY Ns q ´ e

Aps´tsuT N qF pY NtsuT N

q

ı›

›

›

LppP;¨Hγ qds


ż t

0

pt´ sq´η›

›

›Y Ns ´ Y N

tsuT N

›

›

›

LppP;¨Hγ qds

`

„

T

N

ε«

supsPr0,T s

›

›

›F pY N

tsuT Nq

›

›

›

LppP;¨Hγ´η q

ff

T p1´η´εq

p1´ η ´ εq.

(8.125)

Furthermore, Theorem 4.7.6 and Lemma 4.7.7 prove that for all t P r0, T s, ε P


p0, 12` β ´ γq it holds that

„

t

∫0

›

›

›eApt´sq

”

BpY Ns q ´ e

Aps´tsuT N qBpY NtsuT N

q

ı›

›

›

2


12

ď

«

t

∫0pt´ sqp2β´2γq

›

›

›BpY N

s q ´BpYN

tsuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqds

ff12

`

«

t

∫0pt´ sqp2β´2γ´2εq

`

s´ tsuT N˘2ε

›

›

›BpY N

tsuT Nq

›

›

›

2

LppP;¨HSpU,Hβqqds

ff12

ď |B|C0,1pHγ ,HSpU,Hβqq

„

t

∫0pt´ sqp2β´2γq

›

›

›Y Ns ´ Y N

tsuT N

›

›

›

2

LppP;¨Hγ qds

12

`

„

T

N

ε«

supsPr0,T s

›

›

›BpY N

tsuT Nq

›

›

›

LppP;¨HSpU,Hβqq

ff

T p12`β´γ´εq

p1` 2β ´ 2γ ´ 2εq12.

(8.126)

Combining (8.125), (8.126), Corollary 8.2.26, and Lemma 8.2.25 completes the proofof Theorem 8.2.27.

Question 8.2.28 (Convergence speed of exponential Euler approximations). LetT P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq “pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let pWtqtPr0,T s be a cylindri-

cal IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1q ˆRÑ R be a globally Lipschitzcontinuous function, let ξ P H, let X : r0, T s ˆ Ω Ñ H be a mild solution process ofthe SPDE

dXtpxq “B2


(8.127)for x P p0, 1q, t P r0, T s, assume that suptPr0,T sE

“

Xt2H

‰

ă 8, and for every N P N

let Y N : r0, T s ˆΩ Ñ H be a naturally-interpolated exponential Euler approximationfor the SPDE (8.127) with time step size TN.

(i) For which r P R does it holds that there exist a real number C P R such thatfor all N P N it holds that E

“

XT ´ YNT H

‰

ď CN´r?

(ii) For which r P R does it holds that for every p P p0,8q there exist a real numberC P R such that for all N P N it holds that suptPr0,T s

›

›Xt ´ YNt

›

›

LppP;¨Hqď

CN´r?

8.3. NOISE APPROXIMATIONS FOR SPDES 241

8.3 Noise approximations for SPDEs

8.3.1 Noise perturbation estimates

The next result, Corollary 8.3.1, is an immediate consequence of Proposition 7.1.4,Theorem 4.7.6, and Theorem 6.2.32.

Corollary 8.3.1 (Noise perturbation). Assume the setting in Section 7.1.2, let θ Prγ ´ β, 12q, B P C0,1pHγ, HSpU,Hβqq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochasticprocesses which satisfy supsPr0,T s

“

XsLppP;¨Hγ q` XsLppP;¨Hγ q

‰

ă 8, which satisfy

that X is a mild solution of the SPDE


and which satisfy that X is a mild solution of the SPDE

dXt ““

AXt ` F pXtq‰

dt` BpXtq dWt, t P r0, T s, X0 “ ξ. (8.129)

Then

suptPr0,T s

›

›Xt ´ Xt

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a


¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPp0,T q

›

›BpXsq ´ BpXsq›

›

LppP;¨HSpU,Hγ´θqq

ff

ă 8.

(8.130)

Proof. Proposition 7.1.4 ensures that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a


¨?

2 suptPr0,T s

›

›

›

›

„

Xt ´

ż t

0

eApt´sqF pXsq ds´

ż t

0

eApt´sqBpXsq dWs

`

„ż t

0

eApt´sqF pXsq ds`

ż t

0

eApt´sqBpXsq dWs ´ Xt

›

›

›

›

LppP;¨Hγ q

ă 8.

(8.131)


This implies that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a


¨?

2 suptPr0,T s

›

›

›

›

ż t

0

eApt´sq“

BpXsq ´ BpXsq‰

dWs

›

›

›

›

LppP;¨Hγ q

ă 8.

(8.132)

Theorem 6.2.32. and Theorem 4.7.6 hence prove that

suptPr0,T s

›

›X1t ´X

2t

›

›

LppP;¨Hγ q

ď Ep1´ηq„

T 1´η?


`a


¨?

2

«

p pp´1q2

suptPr0,T s

ż t

0

s´2θds

ff12 «

supsPp0,T q

›

›BpXsq ´ BpXsq›

›

LppP;¨HSpU,Hγ´θqq

ff

ă 8.

(8.133)

This completes the proof of Corollary 8.3.1.

8.3.2 Noise approximations for SPDEs

The next result, Corollary 8.3.2, is an immediate consequence from Corollary 8.3.1above.

8.3. NOISE APPROXIMATIONS FOR SPDES 243

Corollary 8.3.2 (Noise discretizations). Assume the setting in Section 7.1.2, letθ P rγ ´ β, 12q, R, R P LpUq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochastic processeswhich satisfy supsPr0,T s

“

XsLppP;¨Hγ q` XsLppP;¨Hγ q

‰

ă 8, which satisfy that X is

a mild solution of the SPDE

dXt “ rAXt ` F pXtqs dt`BpXtqRdWt, t P r0, T s, X0 “ ξ, (8.134)

and which satisfy that X is a mild solution of the SPDE

dXt ““

AXt ` F pXtq‰

dt`BpXtq R dWt, t P r0, T s, X0 “ ξ. (8.135)

Then

suptPr0,T s

›

›Xt ´ Xt

›

›

LppP;¨Hγ qď

«

supvPHγ

BpvqrR ´ RsHSpU,Hγ´θq

maxt1, vHγu

ff

¨ Ep1´ηq„

T 1´η?


`a

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqRLpUq

¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPr0,T s

›

›maxt1, XsHγu›

›

LppP;|¨|q

ff

ă 8.

(8.136)

The next result, Corollary 8.3.3, illustrates how strong convergence rates for noisediscretizations can be obtained.


Corollary 8.3.3. Assume the setting in Section 7.1.2, let θ P rγ ´ β, 12q, r P

r0,8q, pRNqNPN0 Ď LpUq satisfy supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq

maxt1,vHγ uă 8, and

let XN : r0, T s ˆ Ω Ñ Hγ, N P N0, be stochastic processes with the property that@N P N0 : supsPr0,T s XsLppP;¨Hγ q

ă 8 and with the property that for all N P N0 it

holds that XN is a mild solution of the SPDE

dXNt “

“

AXNt ` F pX

Nt q

‰

dt`BpXNt qRN dWt, t P r0, T s, X0 “ ξ. (8.137)

Then

suptPr0,T s

›

›X0t ´X

Nt

›

›

LppP;¨Hγ qď

1

N r

«

supMPN

supvPHγ

M rBpvqrR0 ´RM sHSpU,Hγ´θq

maxt1, vHγu

ff

¨ Ep1´ηq„

T 1´η?


`a

T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqR0LpUq

¨T p12´θq

a

p pp´ 1q?

1´ 2θ

«

suptPr0,T s

supMPN

›

›maxt1, XMs Hγu

›

›

LppP;|¨|q

ff

ă 8.

(8.138)

Proof of Corollary 8.3.3. First, observe that Proposition 7.1.13 ensures that for allM P N0 it holds that

suptPr0,T s

›

›max

1, XMt Hγ

(›

›

LppP;|¨|qď?

2›

›max

1, ξHγ(›

›

LppP;|¨|q

¨ Ep1´ηq„

T 1´η?


`a

T 1´ηppp´ 1q Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq

ă 8.

(8.139)

Next note that the assumption that supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq

maxt1,vHγ uă 8

implies that

supMPN0

Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq“ sup

MPN0

supvPHγ

«

BpvqRMHSpU,Hγ´η2q

max

1, vHγ(

ff

ď supvPHγ

«

BpvqHSpU,Hγ´η2qR0LpUq

max

1, vHγ(

ff

` supMPN0

supvPHγ

«

BpvqrR0 ´RM sHSpU,Hγ´η2q

max

1, vHγ(

ff

ă 8.

(8.140)

8.4. FULL DISCRETIZATIONS FOR SPDES 245

This and (8.139) prove that

supMPN0

suptPr0,T s

›

›max

1, XMt Hγ

(›

›

LppP;|¨|qă 8. (8.141)

This and Corollary 8.3.2 complete the proof of Corollary 8.3.3.

8.4 Full discretizations for SPDEs

8.4.1 Setting

Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letU Ď U be an orthonormal basis of U , let λ : B Ñ R be a function, assume thatDpAq “

v P H :ř

bPB |λb 〈b, v〉H |2ă 8

(

, assume that for all v P DpAq it holds thatAv “

ř

bPB λb 〈b, v〉H b and let pπIqIPPpBq Ď LpHγ´ηq and p$IqIPPpUq Ď LpUq satisfythat for all v P Hγ´η, u P U , I P PpBq, J P PpUq it holds that

πIpvq “ÿ

bPI

〈b, v〉H b and $Jpuq “ÿ

bPJ

〈b, u〉U b. (8.142)

8.4.2 Full-discrete spectral Galerkin exponential Euler methodfor SPDEs

Definition 8.4.1 (Full discrete spectral Galerkin exponential Euler approximations).Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq and letY : t0, 1, . . . , Nu ˆ Ω Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holdsP-a.s. that

Yn`1 “ eATN

˜

Yn ` πIpF pYnqqTN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

. (8.143)

Then we call Y a full-discrete spectral Galerkin exponential Euler approximation forthe SPDE


with time step size TN, spatial approximation I and noise approximation J .


1 function Y = ExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;

10 end

Matlab code 8.1: A Matlab function for simulating a full-discrete spectralGalerkin exponential Euler approximation for the SPDE (7.76).

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = ExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = ExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = ExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.2: A Matlab code for simulating a full-discrete spectral Galerkinexponential Euler approximation for an SPDE of the form (7.76).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 8.1: Result of a call of the Matlab code 8.2.


8.4.3 Full-discrete spectral Galerkin linear-implicit Euler methodfor SPDEs

Definition 8.4.2 (Full-discrete spectral Galerkin linear-implicit Euler approxima-tions). Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq, andlet Y : t0, 1, . . . , Nu ˆΩ Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holdsP-a.s. that

Yn`1 “`

IdH ´TNA˘´1

˜

Yn ` πI`

F pYnq˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

(8.145)

Then we call Y a full-discrete spectral Galerkin linear-implicit Euler approximationfor the SPDE



1 function Y = LinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;

10 end

Matlab code 8.3: A Matlab function for simulating a full-discrete spectralGalerkin linear-implicit Euler approximation of the SPDE (7.76).

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;


6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = LinImpEuler (T, v , f , b , xi ,N,M)9 plot ( Preimage ,Y) ;

10 Y = LinImpEuler (T, v , f , b , xi ,N,M)11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = LinImpEuler (T, v , f , b , xi ,N,M)13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.4: A Matlab code for simulating a full-discrete spectral Galerkinlinear-implicit Euler approximation of the SPDE (7.76).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1



8.4.4 Full-discrete spectral Galerkin nonlinearity-stopped ex-ponential Euler method for SPDEs

Definition 8.4.3 (Full-discrete spectral Galerkin nonlinearity-stopped exponentialEuler approximations). Assume the setting in Section 8.4.1, let N P N, I P PpBq,J P PpUq, α P rγ ´ η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , Nu ˆ Ω ÑπIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ πIpξqand which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.147)

` 1tπIpF pYnqq2HαďNTueA

TN

˜

Yn ` πI`

F pYnq˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

Then we call Y a full-discrete spectral Galerkin nonlinearity-stopped exponential Eu-ler approximation for the SPDE



1 function Y = StopExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;

10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end

Matlab code 8.5: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped exponential Euler approximation for a generalized ver-sion of the SPDE (7.76) with γ P p1

5, 1

4q and α “ 0.

1 clear a l l2 rng ( ’ d e f a u l t ’ )


3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.6: A Matlab code for simulating a full-discrete spectral Galerkinnonlinearity-stopped exponential Euler approximation for a generalized version ofthe SPDE (7.76) with γ P p1

5, 1

4q and α “ 0.

8.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicit Euler method for SPDEs

Definition 8.4.4 (Full-discrete spectral Galerkin nonlinearity-stopped linear-im-plicit Euler approximations). Assume the setting in Section 8.4.1, let N P N, I PPpHq, J P PpUq, α P rγ´ η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , NuˆΩ Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “

πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that

Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.149)

` 1tπIpF pYnqq2HαďNTu

`

IdH ´TNA˘´1

˜

Yn ` πI`

F`

Yn˘˘

TN`

ż pn`1qTN

nTN

πI`

BpYnq$JpdWsq˘

¸

.

Then we call Y a spectral Galerkin nonlinearity-stopped linear-implicit Euler approx-imation for the SPDE




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4



1 function Y = StopLinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;

10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end

Matlab code 8.7: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped linear-implicit Euler approximation for a generalizedversion of the SPDE (7.76) with γ P p1

5, 1

4q and α “ 0.

1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;

10 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f

Matlab code 8.8: A Matlab code for simulating a spectral Galerkin nonlinearity-stopped linear-implicit Euler approximation for a generalized version of theSPDE (7.76) with γ P p1

5, 1

4q and α “ 0.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4


Chapter 9

Weak numerical approximationsfor SPDEs

9.1 An Ito type formula for SPDEs

Most of the following material comes from Da Prato et al. [6].

9.1.1 A setting for mild stochastic calculus

Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq,pH, 〈¨, ¨〉H , ¨Hq, pH, 〈¨, ¨〉H , ¨Hq, and pU, 〈¨, ¨〉U , ¨Uq be separableR-Hilbert spaces

with H Ď H Ď H continuously and densely, let pWtqtPr0,T s be a cylindrical IdU -Wiener process with respect to pFtqtPr0,T s, and let = Ď r0, T s2 be the set given by= “ tpt1, t2q P r0, T s

2 : t1 ă t2u.

255

256 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES

9.1.2 Mild stochastic processes

Definition 9.1.1 (Mild Ito process). Assume the setting in Section 9.1.1, let S PM

`

Bp=q,BpLpH, Hqq˘

satisfy that for all t1, t2, t3 P r0, T s with t1 ă t2 ă t3 it

holds that St2,t3St1,t2 “ St1,t3, and let X : r0, T s ˆ Ω Ñ H, Y : r0, T s ˆ Ω Ñ H, and

Z : r0, T sˆΩ Ñ HSpU, Hq be pFtqtPr0,T s-predictable stochastic processes which satisfy

that for all t P r0, T s it holds P-a.s. thatşt

0Ss,tYsH ` Ss,tZs

2HSpU,Hq ds ă 8 and

which satisfy that for all t P p0, T s it holds P-a.s. that

Xt “ S0,tX0 `

ż t

0

Ss,t Ys ds`

ż t

0

Ss,t Zs dWs. (9.1)

Then we call X a mild Ito process (with evolution family S, mild drift Y , and milddiffusion Z).

Note that if pH, 〈¨, ¨〉H , ¨Hq “ pH, 〈¨, ¨〉H , ¨Hq and if the evolution family

S : = Ñ LpHq satisfies @ pt1, t2q P = : St1,t2 “ IdH in Definition 9.1.1, then themild Ito process (9.1) is nothing else but a “standard” Ito process. (In the followingthe terminology “standard Ito process” instead of simply “Ito process” is used inorder to distinguish more clearly from the above notion of a mild Ito process.) Everystandard Ito process is thus also a mild Ito process. However, a mild Ito processis, in general, not a standard Ito process. The concept of a mild Ito process inDefinition 9.1.1 thus generalizes the concept of a standard Ito process. In concreteexamples of mild Ito processes it will be crucial that the semigroup S : = Ñ LpH, Hqin Definition 9.1.1 depends explicitly on both variables t1 and t2 with pt1, t2q P =

instead of on t2 ´ t1 only (cf. Section 8.2). The assumption that for all t P r0, T sit holds P-a.s. that

şt

0Ss,tYsH ` Ss,tZs

2HSpU,Hq ds ă 8 in Definition 9.1.1 ensures

that both the deterministic and the stochastic integral in (9.1) are well defined. Inthe next step an immediate consequence of Definition 9.1.1 is presented.

Proposition 9.1.2. Assume the setting in Section 9.1.1 and let X : r0, T sˆΩ Ñ H bea mild Ito process with evolution family S : = Ñ LpH, Hq, mild drift Y : r0, T sˆΩ ÑH, and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Then for all t1, t2 P r0, T s witht1 ă t2 it holds P-a.s. that

Xt2 “ St1,t2 Xt1 `

ż t2

t1

Ss,t2 Ys ds`

ż t2

t1

Ss,t2 Zs dWs. (9.2)

9.1. AN ITO TYPE FORMULA FOR SPDES 257

Obviously, equation (9.2) in Proposition 9.1.2 generalizes equation (9.1) in thedefinition of a mild Ito process. Let us complete this subsection on mild Ito processeswith the notion of a certain subclass of mild Ito processes.

Definition 9.1.3 (Mild martingale). Assume the setting in Section 9.1.1 and letX : r0, T sˆΩ Ñ H be a mild Ito process with evolution family S : = Ñ LpH, Hq, milddrift Y : r0, T sˆΩ Ñ H, and mild diffusion Z : r0, T sˆΩ Ñ HSpU, Hq satisfying thatfor all t P r0, T s it holds that E

“

XtH

‰

ă 8 and satisfying that for all t1, t2 P r0, T swith t1 ă t2 it holds P-a.s. that

E“

Xt2

ˇ

ˇFt1‰

“ St1,t2 Xt1 . (9.3)

Then we call X a mild martingale (with respect to the filtration pFtqtPr0,T s and withrespect to the semigroup S).

9.1.3 Mild Ito formula

Theorem 9.1.4 (Mild Ito formula). Assume the setting in Section 9.1.1, let U Ď Ube an orthonormal basis of U , let pV, 〈¨, ¨〉V , ¨V q be a separable R-Hilbert space, letϕ “ pϕpr, xqqrPr0,T s,xPH P C

1,2pr0, T s ˆ H, V q, and let X : r0, T s ˆ Ω Ñ H be a mild

Ito process with evolution family S : = Ñ LpH, Hq, mild drift Y : r0, T s ˆ Ω Ñ H,and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Then for all t0, t P r0, T s with t0 ă tit holds P-a.s. that

ż t

t0

›

›

`

B

Bxϕ˘

ps, Ss,tXsqSs,tYs›

›

V`›

›

`

B

Bxϕqps, Ss,tXsqSs,tZs

›

›

2

HSpU,V qds ă 8, (9.4)

ż t

t0

›

›

`

B

Brϕ˘

ps, Ss,tXsq›

›

V`

`

B2

Bx2ϕ˘

ps, Ss,tXsqLp2qpH,V q Ss,tZs2HSpU,Hq

ds ă 8, (9.5)

ϕpt,Xtq “ ϕpt0, St0,tXt0q `t

∫t0

`

B

Brϕ˘

ps, Ss,tXsq ds`t

∫t0

`

B

Bxϕ˘

ps, Ss,tXsqSs,t Ys ds

`

ż t

t0

`

B

Bxϕ˘

ps, Ss,tXsqSs,t Zs dWs `12

ř

uPU

tş

t0

`

B2

Bx2ϕ˘

ps, Ss,tXsq pSs,tZsu, Ss,tZsuq ds.

(9.6)


Note that (9.4) and (9.5) ensure that the possibly infinite sum and all integralsin (9.6) are well defined. Indeed, equation (9.5) implies that for all t0, t P r0, T s witht0 ă t it holds P-a.s. that

ÿ

uPU

ż t

t0

›

›

`

B2

Bx2ϕ˘

ps, Ss,tXsqpSs,tZsu, Ss,tZsuq›

›

Vds

ď

ż t

t0

›

›

`

B2

Bx2ϕ˘

ps, Ss,tXsq›

›

Lp2qpH,V q

´

ÿ

uPUSs,tZsu

2H

¯

ds

“

ż t

t0

›

›

`

B2

Bx2ϕ˘

ps, Ss,tXsq›

›

Lp2qpH,V q

›

›Ss,tZs›

›

2

HSpU,Hqds ă 8.

(9.7)

Moreover, note that the mild Ito formula (9.6) is independent of the particular choiceof the orthonormal basis U Ď U of U . If the test function pϕpt, xqqtPr0,T s,xPH P

C1,2pr0, T s ˆ H, V q in the mild Ito formula (9.6) does not depend on t P r0, T s, thenthe mild Ito formula in Theorem 9.1.4 reads as follows.

Corollary 9.1.5. Assume the setting in Section 9.1.1, let U Ď U be an orthonormalbasis of U , let pV, 〈¨, ¨〉V , ¨V q be a separable R-Hilbert space, let ϕ P C2pH, V q, and

let X : r0, T s ˆΩ Ñ H be a mild Ito process with evolution family S : = Ñ LpH, Hq,mild drift Y : r0, T s ˆ Ω Ñ H, and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Thenfor all t0, t P r0, T s with t0 ă t it holds P-a.s. that

ż t

t0

ϕ1pSs,tXsqSs,tYsV ` ϕ1pSs,tXsqSs,tZs

2

HSpU,V q ds ă 8, (9.8)

ż t

t0

ϕ2pSs,tXsqLp2qpH,V q Ss,tZs2HSpU,Hq

ds ă 8, (9.9)

ϕpXtq “ ϕpSt0,tXt0q `

ż t

t0

ϕ1pSs,tXsqSs,t Ys ds`

ż t

t0

ϕ1pSs,tXsqSs,t Zs dWs

`1

2

ÿ

uPU

ż t

t0

ϕ2pSs,tXsq pSs,tZsu, Ss,tZsuq ds. (9.10)

Corollary 9.1.5 is an immediate consequence of Theorem 9.1.4. In our proof ofTheorem 9.1.4 below the following elementary result is used.

Exercise 9.1.6. Let Y, Z : r0, T s ˆ Ω Ñ r0,8q be two product measurable stochas-tic processes with the property that for all t P r0, T s it holds that P

“

Yt “ Zt‰

“

P“ şT

0Ys ds ă 8

‰

“ 1. Then P“ şT

0Zs ds ă 8

‰

“ 1.

9.1. AN ITO TYPE FORMULA FOR SPDES 259

In the next step the proof of Theorem 9.1.4 is presented (cf. Conus & Dalang [4],Conus [3], Lindner & Schilling [22], Kovacs, Larsson & Lindgren [19], Debussche &Printemps [8]).

Proof of Theorem 9.1.4. For every t P p0, T s let X t : r0, ts ˆΩ Ñ H be an pFsqsPr0,ts-adapted stochastic processes with continuous sample paths satisfying that for allu P r0, ts, t P p0, T s it holds P-a.s. that

X tu “ S0,tX0 `

ż u

0

Ss,t Ys ds`

ż u

0

Ss,t Zs dWs. (9.11)

Note that the assumption that for all t P r0, T s it holds P-a.s. thatşt

0Ss,tYsH `

Ss,tZs2HSpU,Hq ds ă 8 (see Definition 9.1.1) ensures for all t P p0, T s that X t : rτ, tsˆ

Ω Ñ H in (9.11) is indeed a well defined pFsqsPr0,ts-adapted stochastic processes withcontinuous sample paths. In the next step the continuity of the partial derivativesof ϕ : r0, T s ˆ H Ñ V , the continuity of the sample paths of X t : rτ, ts ˆ Ω Ñ H,t P p0, T s, and again the assumption that for all t P r0, T s it holds P-a.s. thatşt

0Ss,tYsH ` Ss,tZs

2HSpU,Hq ds ă 8 imply that for all t0, t P r0, T s with t0 ă t it

holds that

P„ż t

t0

›

›

`

B

Bxϕ˘

ps, X tsqSs,tYs

›

›

V`›

›

`

B

Bxϕ˘

ps, X tsqSs,tZs

›

›

2

HSpU,V qds ă 8

“ 1 (9.12)

and

P„ż t

t0

›

›

`

B

Brϕ˘

ps, X tsq›

›

V`›

›

`

B2

Bx2ϕ˘

ps, X tsq›

›

Lp2qpH,V q

›

›Ss,tZs›

›

2

HSpU,Hqds ă 8

“ 1.

(9.13)Moreover, the fact that for all t P p0, T s it holds P-a.s. that Xt “ X t

t and the standardIto formula (see Theorem 2.4 in Brzezniak, Van Neerven, Veraar & Weis [1]) provethat for all t0, t P r0, T s with t0 ă t it holds P-a.s. that

ϕpt,Xtq “ ϕpt, X tt q “ ϕpt0, X

tt0q `

ż t

t0

`

B

Brϕ˘

ps, X tsq ds`

ż t

t0

`

B

Bxϕ˘

ps, X tsqSs,t Ys ds

`

ż t

t0

`

B

Bxϕ˘

ps, X tsqSs,t Zs dWs `

1

2

ÿ

uPU

ż t

t0

`

B2

Bx2ϕ˘

ps, X tsq pSs,t Zs u, Ss,t Zs uq ds.

(9.14)


Moreover, note that for all s, t P r0, T s with s ă t it holds P-a.s. that

X ts “ S0,tX0 `

ż s

0

Su,t Yu du`

ż s

0

Su,t Zu dWu

“ Ss,t

ˆ

S0,sX0 `

ż s

0

Su,s Yu du`

ż s

0

Su,s Zu dWu

˙

“ Ss,tXs

(9.15)

Combining (9.15), (9.12), (9.13) with Exercise 9.1.6 implies (9.4) and (9.5). More-over, putting (9.15) into (9.14) proves (9.6). The proof of Theorem 9.1.4 is thuscompleted.

9.2 Solution processes of SPDEs

A direct consequence of Theorem 9.1.4 and Corollary 9.1.5 is the next corollary.

Corollary 9.2.1 (Mild Ito formula for solutions of SPDEs). Assume the setting inSection 7.1.2, let U Ď U be an orthonormal basis of U , and let X : r0, T sˆΩ Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q



Then for all r P p´8,mintγ´η`1, β` 12uq, ϕ P C2pHr, V q, t0, t P r0, T s with t0 ă tit holds P-a.s. that

ż t

t0

›

›ϕ1peApt´sqXsq eApt´sqF pXsq

›

›

Vds ă 8, (9.17)

ż t

t0

›

›ϕ1peApt´sqXsq eApt´sqBpXsq

›

›

2

HSpU,V qds ă 8, (9.18)

ż t

t0

›

›ϕ2peApt´sqXsq›

›

Lp2qpHr,V q

›

›eApt´sqBpXsq›

›

2

HSpU,Hrqds ă 8, (9.19)

ϕpXtq “ ϕpeApt´t0qXt0q `

ż t

t0

ϕ1peApt´sqXsq eApt´sqF pXsq ds

`

ż t

t0

ϕ1peApt´sqXsq eApt´sqBpXsq dWs (9.20)

`1

2

ÿ

uPU

ż t

t0

ϕ2peApt´sqXsq`

eApt´sqBpXsqu, eApt´sqBpXsqu

˘

ds.

9.3. TRANSFORMATIONS OF SEMIGROUPS OF SOLUTIONS OF SPDES 261

9.3 Transformations of semigroups of solutions of

SPDEs

In our definition of a mild solution process in Definition 7.1.1 above we assume thatsuppσP pAqq ă 8. For symmetric diagonal linear operators A : DpAq Ď H Ñ H thecondition suppσP pAqq ă 8 is equivalent to the condition that A is the generator of astrongly continuous semigroup; see Proposition 4.7.2 above for details. In our settingin Section 7.1.2 we impose the more restricitive condition that suppσP pAqq ă 0. Thenext proposition illustrates that the more restricitive condition suppσP pAqq ă 0 doesin a suitable sense not reduce the generality.

Proposition 9.3.1. Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be two separable R-Hilbert spaces, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operator withsuppσP pAqq ă 8, let η P psuppσP pAqq,8q, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to η ´A, let T P r0,8q, α, β, γ, ι P R, O P BpHγq,F P M

`

BpOq,BpHαq˘

, F P M`

BpOq,BpHminpα,γqq˘

, B P M`

BpOq,BpHSpU,Hβqq˘

fulfill that for all v P O it holds that F pvq “ ιv ` F pvq, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let ξ P MpF0,BpOqq, let pWtqtPr0,T s be a cylindrical IdU -Wienerprocess w.r.t. pFtqtPr0,T s, and let X : r0, T s ˆΩ Ñ O be a mild solution of the SPDE


i.e., assume that X is an pFtqtPr0,T s-predictable stochastic process which fulfills thatfor all t P r0, T s it holds P-a.s. that

ż t

0


2HSpU,Hγq ds ă 8 (9.22)

and Xt “ eAtξ `

ż t

0

eApt´sqF pXsq ds`

ż t

0


Then X is a mild solution of the SPDE

dXt ““

pA´ ιqXt ` F pXtq‰

dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (9.24)

i.e., for all t P r0, T s it holds P-a.s. thatż t

0

epA´ιqpt´sqF pXsqHγ ` epA´ιqpt´sqBpXsq

2HSpU,Hγq ds ă 8 (9.25)

and Xt “ epA´ιqtξ `

ż t

0

epA´ιqpt´sqF pXsq ds`

ż t

0

epA´ιqpt´sqBpXsq dWs. (9.26)


Proof of Proposition 9.3.1. First of all, observe that (9.22)–(9.23) imply that for allt P r0, T s it holds P-a.s. that

ż t

0

epA´ιqpt´sq e´ιsF pXsqHγ ` epA´ιqpt´sq e´ιsBpXsq

2HSpU,Hγq ds

“ e´ιtż t

0


2HSpU,Hγq ds ă 8 and

(9.27)

e´ιtXt “ epA´ιqtξ `

ż t

0

epA´ιqpt´sq e´ιsF pXsq ds`

ż t

0

epA´ιqpt´sq e´ιsBpXsq dWs. (9.28)

This implies that the stochastic process pe´ιtXtqtPr0,T s is a mild Ito process withevolution family epA´ιqpt´sq, ps, tq P tpt1, t2q P r0, T s

2 : t1 ă t2u, mild drift e´ιsF pXsq,s P r0, T s, and mild diffusion e´ιsBpXsq, s P r0, T s. Next let ψ, ψ1 : r0, T sˆHγ Ñ Hγ

and ψ2 : r0, T s ˆHγ Ñ LpHγq be the functions with the property that for all pt, xq Pr0, T s ˆHγ, v P Hγ it holds that

ψpt, xq “ eιtx, ψ1pt, xq “B

Btψpt, xq “ ι ψpt, xq, ψ2pt, xq v “

B

Bxψpt, xq v “ ψpt, vq.

(9.29)The mild Ito formula in Theorem 9.1.4 then proves that for all t P r0, T s it holdsP-a.s. that

ż t

0

›

›ψ1

`

s, epA´ιqpt´sq e´ιsXs

˘

loooooooooooooomoooooooooooooon

“ι epA´ιqpt´sqXs

›

›

Hγds ă 8, (9.30)

ż t

0

›

›ψ2

`


˘

epA´ιqpt´sq e´ιsF pXsqloooooooooooooooooooooooooooomoooooooooooooooooooooooooooon

“epA´ιqpt´sqF pXsq

›

›

Hγds ă 8, (9.31)

ż t

0

›

›ψ2

`


˘

epA´ιqpt´sq e´ιsBpXsqloooooooooooooooooooooooooooomoooooooooooooooooooooooooooon

“epA´ιqpt´sqBpXsq

›

›

2

HSpU,Hγqds ă 8, and (9.32)

Xt “ ψ`

t, e´ιtXt

˘

“ ψ`

0, epA´ιqt e´ι0X0

˘

looooooooooomooooooooooon

“epA´ιqtX0

`

ż t

0

epA´ιqpt´sq rιXs ` F pXsqs ds`

ż t

0

epA´ιqpt´sqBpXsq dWs.

(9.33)


9.4. WEAK CONVERGENCE FOR TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES263

9.4 Weak convergence for temporal numerical ap-

proximations for SPDEs

Exercise 9.4.1. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P

p0,8q, N P N, ϕ : L2pBorelp0,1q; |¨|Rq Ñ R satisfy that for all v P L2pBorelp0,1q; |¨|Rq

it holds that ϕpvq “ v2L2pBorelp0,1q;|¨|Rq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic ba-

sis, let pWtqtPr0,T s be a cylindrical IdL2pBorelp0,1q;|¨|Rq-Wiener process w.r.t. pFtqtPr0,T s,

let Y : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be a naturally-interpolated exponential Eulerapproximation for the SPDE


with time step size TN, and let X : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t P r0, T s it holds P-a.s. that

Xt “

ż t

0


Prove that for all r P“

0, 12˘

it holds thatˇ

ˇE“

ϕpXT q‰

É“

ϕpYT q‰ˇ

ˇ ď T r

Nr p12´rq.

We refer to [15] and the references mentioned therein for further weak convergenceresults for SPDEs.

9.5 Weak convergence of Galerkin projections for

SPDEs

The following material comes from Section 2 in Conus et al. [5].

9.5.1 Setting

Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , Üq be separable R-Hilbert spaces, let T P

p0,8q, η P r0, 1q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be acylindrical IdU -Wiener process with respect to pFtqtPr0,T s, let H Ď H be an orthonor-mal basis of H, let λ : HÑ R be a function satisfying supbPH λb ă 0, let A : DpAq ĎH Ñ H be a linear operator such that DpAq “ tv P H :

ř

bPH |λb 〈b, v〉H |2ă 8u and

such that for all v P DpAq it holds that Av “ř

bPH λb 〈b, v〉H b, let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be a family of interpolation spaces associated to Á, and let F P C0,1pH,H´ηq,


B P C0,1pH,HSpU,H´η2qq, ϕ P C2pH,Rq, ξ P L3pP|F0 ; ¨Hq, pπIqIPPpHq Ď LpH´ηq

fulfill that for all v P H, I P PpHq it holds that πIpvq “ř

bPI 〈b, v〉H b.The above assumptions ensure (see Theorem 7.1.9) that there exist an up to

modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆ Ω Ñ Hwhich satisfies suptPr0,T s XtL3pP;¨Hq

ă 8 and which satisfies that for all t P r0, T s itholds P-a.s. that

Xt “ eAtξ `

ż t

0

eApt´sqF pXsq ds`

ż t

0


9.5.2 Weak convergence for spatial spectral Galerkin pro-jections

Proposition 9.5.1. Assume the setting in Section 9.5.1 and let ρ P r0, 1 ´ ηq,I P PpHq. Then

ˇ

ˇE“

ϕpXT q‰

É“

ϕ`

πIpXT q˘‰ˇ

ˇ

ď“

ϕLippH,Rq ` |ϕ1|C0,1pH,LpH,Rqq ` |ϕ

2|C0,1pH,Lp2qpH,Rqq

‰

max

1, suptPr0,T sE“

Xt3H

‰(

¨

«

1

T ρ`T p1´ρ´ηq

“

F LippH,H´ηq ` B2LippH,HSpU,H´η2qq

‰

p1´ ρ´ ηq

ff

“

infbPHzI |λb|‰´ρ

. (9.37)

Proof. Throughout this proof let U Ď U be an orthonormal basis of U and letBb P CpH,H´η2q, b P U, be the functions with the property that for all v P H,b P U it holds that Bbpvq “ Bpvq b. Then observe that the mild Ito formula inCorollary 9.2.1 yields that

E“

ϕpXT q‰

É“

ϕpπIpXT qq‰

“ E“

ϕpeAT ξq‰

É“

ϕpeATπIpξqq‰

`

ż T

0

E“

ϕ1peApT´tqXtq eApT´tqF pXtq

‰

dt

´

ż T

0

E“

ϕ1peApT´tqπIpXtqq eApT´tqπIF pXtq

‰

dt

`1

2

ÿ

bPU

ż T

0

E“

ϕ2peApT´tqXtqpeApT´tqBb

pXtq, eApT´tqBb

pXtqq‰

dt

´1

2

ÿ

bPU

ż T

0

E“

ϕ2peApT´tqπIpXtqqpeApT´tqπIB

bpXtq, e

ApT´tqπIBbpXtqq

‰

dt.

(9.38)

9.5. WEAK CONVERGENCE OF GALERKIN PROJECTIONS FOR SPDES265

Next observe that the fact that for all r P r0,8q it holds that suptPp0,8q›

›p´tAqreAt›

›

LpHqď

supxPp0,8q“

xr

ex

‰

ď“

re

‰r(see Theorem 4.7.6) implies that

ˇ

ˇE“

ϕpeAT ξq‰

´E“

ϕpeATπIpξqq‰ˇ

ˇ ď|ϕ|C0,1pH,Rq ξL1pP;¨Hq

πHzILpH,H´ρq

T ρ. (9.39)

Inequality (9.39) provides us a bound for the first difference on the right hand sideof (9.38). In the next step we bound the second difference on the right hand side of(9.38). For this observe that for all x P H, t P r0, T q it holds that

ˇ

ˇ

“

ϕ1peApT´tqxq ´ ϕ1peApT´tqπIpxqq‰

eApT´tqF pxqˇ

ˇ

ď|ϕ1|C0,1pH,LpH,Rq πHzILpH,H´ρq xH F pxqH´η

pT ´ tqpρ`ηq(9.40)

andˇ

ˇϕ1peApT´tqπIpxqq`

rIdH ´πIs eApT´tqF pxq

˘ˇ

ˇ

ď|ϕ|C0,1pH,Rq πHzILpH,H´ρq F pxqH´η

pT ´ tqpρ`ηq.

(9.41)

Combining (9.40) and (9.41) proves that

ˇ

ˇ

ˇ

ˇ

ż T

0

E“

ϕ1peApT´tqXtq eApT´tqF pXtq

‰

dt´

ż T

0

E“

ϕ1peApT´tqπIpXtqq eApT´tqπIF pXtq

‰

dt

ˇ

ˇ

ˇ

ˇ

ďT p1´ρ´ηq suptPr0,T sE

“

XtH F pXtqH´η |ϕ1|C0,1pH,LpH,Rqq`F pXtqH´η |ϕ|C0,1pH,Rq

‰

πHzILpH,H´ρq

p1´ρ´ηq

ď

T p1´ρ´ηq r|ϕ1|C0,1pH,LpH,Rqq`|ϕ|C0,1pH,Rqs suptPr0,T s

maxtErXtH F pXtqH´η s,ErF pXtqH´η su πHzILpH,H´ρq

p1´ρ´ηq

ďT p1´ρ´ηq r|ϕ1|C0,1pH,LpH,Rqq`|ϕ|C0,1pH,Rqs F LippH,H´ηq

maxt1,suptPr0,T sErXt2H su πHzILpH,H´ρq

p1´ρ´ηq.

(9.42)

Inequality (9.42) provides us a bound for the second difference on the right hand sideof (9.38). Next we bound the third difference on the right hand side of (9.38). Tothis end note that for all x P H, t P r0, T q it holds that

ˇ

ˇ

ˇ

ˇ

ř

bPU

“

ϕ2peApT´tqxq ´ ϕ2peApT´tqπIpxqq‰

peApT´tqBbpxq, eApT´tqBbpxqq

ˇ

ˇ

ˇ

ˇ

ď|ϕ2|C0,1pH,Lp2qpH,Rqq Bpxq

2HSpU,H´η2q

xH πHzILpH,H´ρq

pT ´ tqpρ`ηq

(9.43)


andˇ

ˇ

ˇ

ˇ

ř

bPUϕ2peApT´tqπIpxqqprIdH `πIse

ApT´tqBbpxq, rIdH ´πIseApT´tqBbpxqq

ˇ

ˇ

ˇ

ˇ

ď2 |ϕ1|C0,1pH,LpH,Rqq Bpxq

2HSpU,H´η2q

πHzILpH,H´ρq

pT ´ tqpρ`ηq.

(9.44)

Combining (9.43) and (9.44) proves that

ˇ

ˇ

ˇ

ˇ

ˇ

1

2

ÿ

bPU

ż T

0

E“

ϕ2peApT´tqXtqpeApT´tqBb

pXtq, eApT´tqBb

pXtqq‰

dt

´1

2

ÿ

bPU

ż T

0

E“

ϕ2peApT´tqπIpXtqqpeApT´tqπIB

bpXtq, e

ApT´tqπIBbpXtqq

‰

dt

ˇ

ˇ

ˇ

ˇ

ˇ

ďT p1´ρ´ηq πHzILpH,H´ρq

“

|ϕ2|C0,1pH,Lp2qpH,Rqq ` |ϕ1|C0,1pH,LpH,Rqq

‰

p1´ ρ´ ηq

¨

«

suptPr0,T s

max

E“

XtHBpXtq2HSpU,H´η2q

‰

,E“

BpXtq2HSpU,H´η2q

‰(

ff

ďT p1´ρ´ηq πHzILpH,H´ρq

“

|ϕ2|C0,1pH,Lp2qpH,Rqq ` |ϕ1|C0,1pH,LpH,Rqq

‰

p1´ ρ´ ηq

¨ B2LippH,HSpU,H´η2qqmax

!

1, suptPr0,T s

E“

Xt3H

‰

)

.

(9.45)

Combining (9.38), (9.39), (9.42), and (9.45) completes the proof of Proposition 9.5.1.

Question 9.5.2. Assume the setting in Section 9.5.1, assume that ϕ P C3pH,Rq,and assume that ϕ has globally bounded derivatives. For which ρ P R does it holdthat there exists a real number C P R such that for all I P PpHq it holds that

ˇ

ˇE“

ϕpXT q‰

´E“

ϕ`

πIpXT q˘‰ˇ

ˇ ď C“

infbPHzI |λb|‰´ρ

. (9.46)

9.5. WEAK CONVERGENCE OF GALERKIN PROJECTIONS FOR SPDES267

Question 9.5.3 (Weak convergence rates of spectral Galerkin approximations).Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq, let pWtqtPr0,T s be a cylin-

drical IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1qˆRÑ R be a globally Lipschitzcontinuous function, let ξ P H, let X : r0, T s ˆ Ω Ñ H be a mild solution process ofthe SPDE

dXtpxq “B2


(9.47)for x P p0, 1q, t P r0, T s, and let πN P LpHq, N P N, satisfy that for all N P N, v P Hit holds that

πNpvq “Nÿ

n“1

2 sinpnπp¨qq

ż 1

0

sinpnπxq vpxq dx. (9.48)

(i) For which r P R does it holds that there exists a real number C P r0,8q suchthat for all N P N it holds that ErXT ´ πNpXT qHs ď CN´r?

(ii) For which r P R does it holds that for every ϕ P C3pH,Rq with globally boundedderivatives there exists a real number C P r0,8q such that for all N P N it holdsthat

“

E“

ϕpXT q‰

´E“

ϕpπNpXT qq‰ˇ

ˇ ď CN´r?

Chapter 10

Additional material

10.1 Egorov’s theorem on almost uniform conver-

gence

See, e.g., Wikipedia http://en.wikipedia.org/wiki/Egorov%27s_theorem#Generalizationsfor the next results and their proofs.

10.1.1 General measure spaces

10.1.1.1 Almost sure convergence

Proposition 10.1.1 (A characterization for almost sure convergence). Let pΩ,F , µqbe a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following two statements areequivalent:

(i) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0.

(ii) It holds for all ε P p0,8q that µ`

XnPNtsupmPNXrn,8q dpfm, f0q ą εu˘

“ 0.

Proof of Proposition 10.1.1. Observe that

µpΩz tlim supnÑ8 dpfn, f0q “ 0uq

“ µ´

Ω z

@ k P N : Dn P N : @m P NX rn,8q : dpfm, f0q ď1k

(

¯

“ µ´

Ω z

@ k P N : Dn P N : supmPNXrn,8q dpfm, f0q ď1k

(

¯

“ µ´

YkPN XnPN

supmPNXrn,8q dpfm, f0q ą1k

(

¯

.

(10.1)

269

http://en.wikipedia.org/wiki/Egorov%27s_theorem#Generalizations

270 CHAPTER 10. ADDITIONAL MATERIAL

This proves that it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 if and only if for all k P Nit holds that µ

`

XnPNtsupmPNXrn,8q dpfm, f0q ą1ku˘

“ 0. This completes the proof ofProposition 10.1.1.

Corollary 10.1.2 (Almost sure convergence). Let pΩ,F , µq be a measure space, letpE, dq be a metric space, and let fn : Ω Ñ E, n P N0, be strongly F/pE, dq-measurablefunctions with the property that it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0. Then itholds for all ε P p0,8q, A P F with µpAq ă 8 that

limnÑ8

µ`

AX tsupmPNXrn,8q dpfm, f0q ą εu˘

“ 0. (10.2)

Proof of Corollary 10.1.2. Proposition 10.1.1 implies that for all ε P p0,8q it holdsthat µ

`


“ 0. This and continuity from above, inparticular, ensure that for all ε P p0,8q, A P F with µpAq ă 8 it holds that

0 “ µ`

AX`

XnPN tsupmPNXrn,8q dpfm, f0q ą εu˘˘

“ µ`

XnPN`

AX tsupmPNXrn,8q dpfm, f0q ą εu˘˘

“ limnÑ8

µ`


.

(10.3)


Proposition 10.1.3 (A modified verison of the Sevirini-Egorov theorem). Let pΩ,F , µqbe a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following two statements areequivalent:

(i) It holds for all ε P p0,8q, A P F that there exist a set B P pA \ Fq such thatµpAzBq ă ε and limnÑ8 supωPB dpfnpωq, f0pωqq “ 0.

(ii) It holds for all ε P p0,8q, A P F with µpAq ă 8 that

limnÑ8

µ`


“ 0. (10.4)

Proof of Proposition 10.1.3. Proposition 10.1.3 implies that for all ε P p0,8q it holdsthat µ

`


“ 0. This and continuity from above, inparticular, ensure that for all ε P p0,8q, A P F with µpAq ă 8 it holds that

0 “ µ`

AX`

XnPN tsupmPNXrn,8q dpfm, f0q ą εu˘˘

“ µ`

XnPN`

AX tsupmPNXrn,8q dpfm, f0q ą εu˘˘

“ limnÑ8

µ`


.

(10.5)


10.1. EGOROV’S THEOREM ON ALMOST UNIFORM CONVERGENCE 271

10.1.1.2 Luzin uniform-type convergence

Lemma 10.1.4 (Luzin uniform-type convergence implies almost sure convergence).Let pΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E,n P N0, be strongly F/pE, dq-measurable functions with the property that there existsets Ak P F , k P N, such that @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 andµ`

ΩzpYkPNAkq˘

“ 0. Then it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0.

Proof of Lemma 10.1.4. Observe that the assumption that for all k P N it holds thatlimnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 implies that

pYkPNAkq Ď!

ω P Ω: limnÑ8

dpfnpωq, f0pωqq “ 0)

. (10.6)

Combining this with the assumption that µ`

ΩzpYkPNAkq˘

“ 0 completes the proofof Lemma 10.1.4.

10.1.1.3 Almost uniform convergence

Lemma 10.1.5 (Almost uniform convergence implies Luzin uniform-type conver-gence). Let pΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω ÑE, n P N0, be strongly F/pE, dq-measurable functions with the property that for allε P p0,8q there exists a set A P F such that limnÑ8 supωPA dpfnpωq, f0pωqq “ 0and µpΩzAq ă ε. Then it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 and it holdsthat there exist sets Ak P F , k P N, such that µ

`

ΩzpYkPNAkq˘

“ 0 and @ k PN : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0.

Proof of Lemma 10.1.5. By assumption that there exists sets Ak P F , k P N, suchthat for all k P N it holds that

limnÑ8

supωPAk

dpfnpωq, f0pωqq “ 0 and µpΩzAkq ă1k. (10.7)

This ensures that

µpΩz pYkPNAkqq “ µpXkPN pΩzAkqq ě limkÑ8

µpΩzAkq “ 0. (10.8)

Combining this and (10.7) with Lemma 10.1.4 completes the proof of Lemma 10.1.5.


10.1.2 Finite measure spaces

Theorem 10.1.6 (Severini-Egorov-Luzin theorem for finite measure spaces). LetpΩ,F , µq be a finite measure space, let pE, dq be a metric space, and let fn : Ω Ñ E,n P N0, be strongly F/pE, dq-measurable functions. Then the following five state-ments are equivalent:

(i) It holds for all ε P p0,8q that there exists a set A P F such that µpΩzAq ă εand limnÑ8 supωPA dpfnpωq, f0pωqq “ 0 (almost uniform convergence).

(ii) It holds that there exist sets Ak P F , k P N, such that µ`

ΩzpYkPNAkq˘

“ 0,@ k P N : µpAkq ă 8, and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzinfinite-uniform-type convergence).

(iii) It holds that there exist sets Ak P F , k P N, such that µ`

ΩzpYkPNAkq˘

“ 0and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzin uniform-type conver-gence).

(iv) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almost sure convergence).

(v) It holds for all ε P p0,8q that limnÑ8 µ`

supmPNXrn,8q dpfm, f0q ą ε˘

“ 0.

Proof of Theorem 10.1.6. Lemma 10.1.5 ensures that piq ñ piiq. The statementthat piiq ô piiiq is clear. Lemma 10.1.4 implies that piiiq ñ pivq. Proposition 10.1.1together with the fact that the measure µ is continuous from above establishes thatpivq ñ pvq. It thus remains to prove that pvq ñ piq. For this let ε P p0,8q bearbitrary and assume that for all k P N it holds that

limnÑ8

µ`

supmPNXrn,8q dpfm, f0q ą1k

˘

“ 0. (10.9)

Observe that (10.9) shows that there exists a sequence nk P N, k P N, such that forall k P N it holds that

µ`

supmPNXrnk,8q dpfm, f0q ą1k

˘

“ µ`

Ω z

supmPNXrnk,8q dpfm, f0q ď1k

(˘

ă ε2k.

(10.10)Next let A Ď Ω be the set given by

A “ XkPN


(

“

@ k P N : @m P NX rnk,8q : dpfm, f0q ď1k

( (10.11)

and note that (10.10) shows that

µpΩzAq ďř8

k“1 µ`

Ωz


(˘

ďř8

k“1ε

2k“ ε. (10.12)

This proves that pvq ñ piq. The proof of Theorem 10.1.6 is thus completed.

10.1. EGOROV’S THEOREM ON ALMOST UNIFORM CONVERGENCE 273

10.1.3 Sigma-finite measure spaces

Corollary 10.1.7 (Severini-Egorov-Luzin theorem for sigma-finite measure spaces).Let pΩ,F , µq be a sigma-finite measure space, let pE, dq be a metric space, and letfn : Ω Ñ E, n P N0, be strongly F/pE, dq-measurable functions. Then the followingfive statements are equivalent:


(ii) It holds that there exist sets Ak P F , k P N, such that µ`

ΩzpYkPNAkq˘

“ 0,@ k P N : µpAkq ă 8, and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzinfinite-uniform-type convergence).

(iii) It holds that there exist sets Ak P F , k P N, such that µ`

ΩzpYkPNAkq˘

“ 0and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzin uniform-type conver-gence).

(iv) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almost sure convergence).

(v) It holds for all ε P p0,8q, A P F with µpAq ă 8 that

limnÑ8

µ`

AX

supmPNXrn,8q dpfm, f0q ą ε(˘

“ 0. (10.13)

Proof of Corollary 10.1.7. First, we observe that clearly piq ñ piiq. Next we notethat Lemma 10.1.4 ensures that piiq ñ piiiq. Moreover, we observe that Corl-lary 10.1.2 shows that piiiq ñ pivq. It thus remains to prove that pivq ñ piq. Wethus assume in the following that for all ε P p0,8q, A P F with µpAq ă 8 it holdsthat

limnÑ8

µ`

AX


“ 0. (10.14)

Moreover, we observe that the assumption that µ is sigma-finite implies that thereexists a sequence Ωk P F , k P N, such that Ω “ YkPNΩk and @ k P N : µpΩkq ă 8.Combining this with (10.14) shows that for all k P N, ε P p0,8q it holds that

limnÑ8

µ|Ωk\F`

supmPNXrn,8q dpfm|Ωk , f0|Ωkq ą ε(˘

“ 0. (10.15)

Theorem 10.1.6 hence proves that for all k P N there exist sets Ak,l P pΩk \ Fq,l P N, such that

µ|Ωk\F`

ΩkzpYlPNAk,lq˘

“ 0 and @ l P N : limnÑ8

supωPAk,l

dpfnpωq, f0pωqq “ 0.

(10.16)


This shows that

µpΩ z pYk,lPNAk,lqq “ µpΩ z pZkPN rYlPNAk,lsqq

“ µpZkPN rΩk z pYlPNAk,lqsq “8ÿ

k“1

µpΩk z pYlPNAk,lqq “ 0.(10.17)

Combining this and (10.16) proves that pivq ñ piq. The proof of Corollary 10.1.7 isthus completed.

10.1.4 General measure spaces

Theorem 10.1.8 (Severini-Egorov-Luzin theorem for general measure spaces). LetpΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following five statements areequivalent:


(ii) For all A P F with µpAq ă 8 it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almostsure convergence).

(iii) It holds for all ε P p0,8q, A P F with µpAq ă 8 that

limnÑ8

µ`

AX


“ 0. (10.18)

10.2 Fast convergence in probability

Let pE, dEq be a metric space and let an P E, n P N0, be a sequence of elements ofE with the property that

8ÿ

n“1

dEpan, a0q ă 8. (10.19)

Then it holds, in particular, that limnÑ8 an “ a0, i.e., it holds that panqnPN convergesto a0 in pE, dEq. The property (10.19) is sometimes referred as fast convergenceof panqnPN to a0 in pE, dEq. Fast convergence in probability ensures almost sureconvergence. This is the subject of the next lemma.

10.3. DINI’S THEOREMON POINTWISE CONVERGENCEOF CONTINUOUS FUNCTIONS275

Lemma 10.2.1 (Fast convergence implies almost sure convergence). Let pΩ,F ,Pq bea probability space, let pE, dEq be a metric space, let Xn : Ω Ñ E, n P N0, be stronglyF/pE, dEq-measurable mappings with the property that

ř8

n“1E“

min

1, dEpXn, X0q(‰

ă

8. Then it holds P-a.s. that limnÑ8Xn “ X0.

Proof of Lemma 10.2.1. First of all, observe that the Markov inequality and theassumption that

ř8

n“1E“

min

1, dEpXn, X0q(‰

ă 8 ensure that for all ε P p0, 1s itholds that

8ÿ

n“1

P`

dEpXn, X0q ě ε(˘

“

8ÿ

n“1

P`

mint1, dEpXn, X0qu ě ε(˘

ď

8ÿ

n“1

«

E“

mint1, dEpXn, X0qu‰

ε

ff

ă 8.

(10.20)

The lemma of Borel-Cantelli hence implies that for all ε P p0, 1s it holds that

Pˆ

lim supnÑ8

tdEpXn, X0q ě εu

˙

“ 0. (10.21)

This proves that for all ε P p0, 1s it holds that

PptDn P N : @m P NX rn,8q : dEpXm, X0q ă εuq “ P´

lim infnÑ8

tdEpXn, X0q ă εu¯

“ 1.

(10.22)Continuity from above of the probability measure P hence shows that

Ppt@ ε P p0,8q : Dn P N : @m P NX rn,8q : dEpXm, X0q ă εuq

“ P`

XεPp0,8q tDn P N : @m P NX rn,8q : dEpXm, X0q ă εu˘

“ limεŒ0

PptDn P N : @m P NX rn,8q : dEpXm, X0q ă εuq “ 1.(10.23)


10.3 Dini’s theorem on pointwise convergence of

continuous functions

See, e.g., wikipedia for the next results and their proofs.

Proposition 10.3.1 (Dini’s theorem). Let pX,X q be a compact topological space andlet fn : X Ñ r0,8q, n P N, be non-increasing continuous functions with the propertythat limnÑ8 fnpxq “ 0. Then limnÑ8 psupxPX fnpxqq “ 0.


Proof of Proposition 10.3.1. Throughout this proof let Xn,ε Ď X, n P N, ε P p0,8q,be the sets with the property that for all n P N, ε P p0,8q it holds that

Xn,ε “ tx P X : fnpxq ă εu “ f´1n pp´8, εqq . (10.24)

The assumption that @x P X : limnÑ8 fnpxq “ 0 implies that for all ε P p0,8q itholds that

YnPNXn,ε “ X. (10.25)

In the next step we observe that the assumption that fn : X Ñ R, n P N0, arecontinuous functions proves that for all n P N, ε P p0,8q it holds that Xn,ε P X .Combining this with (10.25) and the assumption that X is a compact set shows thatfor all ε P p0,8q there exists a natural number m P N such that

Ymn“1Xn,ε “ X. (10.26)

In addition, we observe that the asumption that @x P X, n P N : fn`1pxq ď fnpxqensures that for all ε P p0,8q, n P N it holds that Xn,ε Ď Xn`1,ε. This and (10.26)prove that for all ε P p0,8q there exists a natural number m P N such that

X “ Ymn“1Xn,ε “ Xm,ε “ X

8n“mXn,ε. (10.27)


Theorem 10.3.2 (Dini’s theorem). Let pX,X q be a compact topological space, letpE, dq be a metric space, and let fn : X Ñ E, n P N0, be continuous functionswith the property that @x P X, n P N : d

`

f0pxq, fn`1pxq˘

ď d`

f0pxq, fnpxq˘

andlimmÑ8 fmpxq “ f0pxq. Then

limnÑ8

“

supxPX d`

f0pxq, fnpxq˘‰

“ 0. (10.28)

Proof of Theorem 10.3.2. Throughout this proof let gn : X Ñ r0,8q, n P N, be thefunctions with the property that for all n P N, x P X it holds that

gnpxq “ d`

fnpxq, f0pxq˘

. (10.29)

Proposition 10.3.1 then proves that limnÑ8

`

supxPX gnpxq˘

“ 0. The proof of Theo-rem 10.3.2 is thus completed.


10.3.1 On the compactness of the argument space

The next example shows that the assumption in Theorem 10.3.2 that X is a compactset can not be avoided.

Example 10.3.3. Let fn : r0,8q Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “

xn

and f0pxq “ 0. Then

• it holds for all n P N0 that fn P Cpr0,8q,Rq,

• it holds for all x P r0,8q, n P N that

|fn`1pxq ´ f0pxq| “x

pn` 1qďx

n“ |fnpxq ´ f0pxq| , (10.30)

• it holds for all x P r0,8q that limnÑ8 fnpxq “ f0pxq, and

• it holds that

limnÑ8

«

supxPr0,8q

|fnpxq ´ f0pxq|

ff

“ limnÑ8

«

supxPr0,8q

xn

ff

“ 8. (10.31)

10.3.2 On the monotonicity of the approximating functions

The next example shows that the assumption in Theorem 10.3.2 that it holds forevery x P X that the sequence dpf0pxq, fnpxqq, n P N, is non-increasing can not be


avoided.

Example 10.3.4. Let gr,ε : R Ñ r0, 1s, r P R, ε P p0,8q, be the functions with theproperty that for all r P R, ε P p0,8q it holds that

gr,εpxq “

$

’

’

&

’

’

%

0 : x P Rzpr ´ ε, r ` εq

1εpx´ pr ´ εqq : x P rr ´ ε, rs

1εppr ` εq ´ xq : x P rr, r ` εs

(10.32)

and let fn : r0, 1s Ñ R, n P N0, be the functions with the property that for all n P N,x P r0, 1s it holds that fnpxq “ gn´1,2ńpxq and f0pxq “ 0. Then

• it holds for all n P N0 that fn P Cpr0, 1s,Rq,

• it holds for all x P r0, 1s that limnÑ8 fnpxq “ f0pxq, and

• it holds that

limnÑ8

«

supxPr0,1s

|fnpxq ´ f0pxq|

ff

“ limnÑ8

«

supxPr0,1s

gn´1,2ńpxq

ff

“ 1. (10.33)

10.3.3 On the continuity of the approximating functions

The next example shows that the assumption in Theorem 10.3.2 that the functionsfn, n P N, are continuous can not be avoided.

Example 10.3.5. Let fn : r0, 1s Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “ 1

R

p0,2ńqpxq and f0pxq “ 0. Then

• it holds that f0 P Cpr0, 1s,Rq,

• it holds for all x P r0, 1s, n P N that

|fn`1pxq ´ f0pxq| “ 1R

p0,2´pn`1qqpxq ď 1Rp0,2ńqpxq “ |fnpxq ´ f0pxq| , (10.34)


• it holds that

limnÑ8

«

supxPr0,1s

|fnpxq ´ f0pxq|

ff

“ limnÑ8

«

supxPr0,1s

1p0,2ńqpxq

ff

“ 1. (10.35)


10.3.4 On the continuity of the limit function

The next example shows that the assumption in Theorem 10.3.2 that the functionf0 is continuous can not be avoided.

Example 10.3.6. Let fn : r0, 1s Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “ xn and f0pxq “ 1

R

t1upxq. Then

• it holds for all n P N that fn P Cpr0, 1s,Rq,

• it holds for all x P r0, 1s, n P N that

|fn`1pxq ´ f0pxq| “ 1R

r0,1qpxq ¨ xpn`1q

ď 1Rr0,1qpxq ¨ xn“ |fnpxq ´ f0pxq| , (10.36)


• it holds that

limnÑ8

«

supxPr0,1s

|fnpxq ´ f0pxq|

ff

“ limnÑ8

«

supxPr0,1q

xn

ff

“ 1. (10.37)

Chapter 11

Solutions to selected exercises

11.1 Chapter 2

11.1.1 Solution to Exercise 2.2.6

Lemma 11.1.1. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space andlet f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if it holdsfor all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.

Proof of Lemma 11.1.1. First of all, recall that every every ϕ P CpE,Rq is BpEq/BpRq-measurable. This shows that if f is F/BpEq-measurable, then it holds for everyϕ P CpE,Rq that the composition ϕ ˝ f is F/BpRq-measurable. It thus remainsto prove that if for every ϕ P CpE,Rq it holds that ϕ ˝ f is F/BpRq-measurable,then f is F/BpEq-measurable. This, in turn, is an immediate consequence fromProposition 2.2.5. The proof of Lemma 11.1.1 is thus completed.


Lemma 11.1.2 (Example of a measurable but not strongly measurable function).It holds that the function IdL2p#R;|¨|

Rq is B

`

L2p#R; |¨|Rq˘

/B`

L2p#R; |¨|Rq˘

-measurable

but not B`

L2p#R; |¨|Rq˘

/`

L2p#R; |¨|Rq, ¨L2p#R;|¨|

Rq

˘

-measurable.

Proof of Lemma 11.1.2. Clearly, IdL2p#R;|¨|Rq is continuous. Hence, IdL2p#R;|¨|

Rq is

also B`

L2p#R; |¨|Rq˘

/B`

L2p#R; |¨|Rq˘

-measurable. Next note that Example 3.3.8shows that L2p#R; |¨|

Rq is not separable. Combining this with the fact that

im`

IdL2p#R;|¨|Rq

˘

“ L2p#R; |¨|

Rq (11.1)

281

282 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES

completes the proof of Lemma 11.1.2.

11.2 Chapter 3


Lemma 11.2.1. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let

A : DpAq Ď H Ñ H be a diagonal linear operator. Then DpAqH“ H.

Proof of Lemma 11.2.1. The assumption that A : DpAq Ď H Ñ H is a diagonallinear operator ensures that there exists an orthonormal basis B Ď H of H and afunction λ : BÑ K such that

DpAq “

#

v P H :ÿ

bPB

|λb|2|〈b, v〉H |

2ă 8

+

(11.2)


Av “ÿ

bPB

λb 〈b, v〉H b. (11.3)

Equation (11.2) implies thatB Ď DpAq. (11.4)

Moreover, the fact that B is an orthonormal basis ensures that BH“ H. This and

(11.4) prove that DpAqH“ H. The proof of Lemma 11.2.1 is thus completed.

11.3. CHAPTER 7 283

11.3 Chapter 7





Then for all r P“

γ,mint1 ` γ ´ η, 12 ` βu˘

, ε P`

0,mint1 ` γ ´ η, 12 ` βu ´ r˘

itholds that

supt1,t2Pr0,T s,t1‰t2

¨

˝

›

›

`

Xt1 ´ et1AX0

˘

´`

Xt2 ´ et2AX0

˘›

›

LppP;¨Hr q

|t1 ´ t2|ε

˛

‚

ď

«

supsPr0,T s

F pXsqLppP;¨Hγ´ηq

ff

2T p1`γ´η´r´εq

p1` γ ´ η ´ ε´ rq

`

«

supsPr0,T s

BpXsqLppP;¨HSpU,Hβq

ff

a

p pp´ 1qT p12`β´r´εq

p1` 2β ´ 2r ´ 2εq12ă 8.

(11.6)

Proof of Proposition 11.3.1. First of all, note that Theorem 4.7.6 and Lemma 4.7.7imply that for all r P rγ, 1` γ ´ ηq, ε P p0, 1` γ ´ η ´ rq, t1, t2 P r0, T s with t1 ă t2


it holds that›

›

›

›

ż t1

0

ept1´sqAF pXsq ds´

ż t2

0

ept2´sqAF pXsq ds

›

›

›

›

LppP;¨Hr q

ď

ż t2

t1

›

›ept2´sqAF pXsq›

›

LppP;Hrqds`

ż t1

0

›

›

`

ept1´sqA ´ ept2´sqA˘

F pXsq›

›

LppP;¨Hr qds

ď

«

supsPr0,T s


ff

¨

„ż t2

t1

pt2 ´ sqγ´η´r ds`

ż t1

0

pt1 ´ sqγ´η´r´ε

pt2 ´ t1qε ds

“

«

supsPr0,T s


ff«

pt2 ´ t1qp1`γ´η´rq

p1` γ ´ η ´ rq`pt2 ´ t1q

εpt1q

p1`γ´η´r´εq

p1` γ ´ η ´ r ´ εq

ff

ď

«

supsPr0,T s


ff

„

2T p1`γ´η´r´εq pt2 ´ t1qε

p1` γ ´ η ´ ε´ rq

.

(11.7)

Moreover, observe that Theorem 4.7.6 and Lemma 4.7.7 ensure that for all r Prγ, 12` βq, ε P p0, 12` β ´ rq, t1, t2 P r0, T s with t1 ă t2 it holds that

›

›

›

›

ż t1

0

ept1´sqABpXsq dWs ´

ż t2

0

ept2´sqABpXsq dWs

›

›

›

›

2

LppP;¨Hr q

ďp pp´1q

2

ż t2

t1

›

›ept2´sqABpXsq›

›

2

LppP;¨HSpU,Hrqqds

`p pp´1q

2

ż t1

0

›

›

`

ept1´sqA ´ ept2´sqA˘

BpXsq›

›

2

LppP;¨HSpU,Hrqqds

ďp pp´1q

2

«

supsPr0,T s


ff2

¨

„ż t2

t1

pt2 ´ sqp2β´2rq ds`

ż t1

0

pt1 ´ sqp2β´2r´2εq

pt2 ´ t1q2ε ds

ďp pp´1q

2

«

supsPr0,T s


ff22 pt2 ´ t1q

2εpt2q

p1`2β´2r´2εq

p1` 2β ´ 2r ´ 2εq.

(11.8)

Combining (11.7) and (11.8) completes the proof of Proposition 11.3.1.

11.3. CHAPTER 7 285


Corollary 11.3.2. Assume the setting in Subsection 7.1.2, let δ P rγ,8q, assumethat ξ P LppP; ¨Hδq, and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications uniquepFtqtPr0,T s-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q

ă 8

and which is a mild solution of the SPDE


Then for all r P“

γ,mint1 ` γ ´ η, 12 ` βu˘

, ε P`

0,mint1 ` γ ´ η, 12 ` βu ´ r˘

itholds that


¨

˝


Xt1 ´Xt2LppP;¨Hr q

|t1 ´ t2|ε

˛

‚

ď X0LppP;¨Hmintδ,r`εuq`

«

supsPr0,T s


ff

2T p1`γ´η´mintδ,r`εuq

p1` γ ´ η ´ ε´ rq

`

«

supsPr0,T s


ff ?2T p12`β´mintδ,r`εuq

p1` 2β ´ 2r ´ 2εq12ă 8.

(11.10)

Proof of Corollary 11.3.2. Note that Lemma 4.7.7 ensures that for all r P“

γ,mint1`γ ´ η, 12` βu

˘

, ε P`

0,mint1` γ ´ η, 12` βu ´ r˘

it holds that


¨

˝


›

›et1AX0 ´ et2AX0

›

›

LppP;¨Hr q

|t1 ´ t2|ε

˛

‚

ď supt1,t2Pr0,T s,t1‰t2

˜

|mintt1,t2u|maxtr`ε´δ,0u

pÁqr´mintδ,r`εupet1Aét2AqLpHq X0LppP;¨Hmintδ,r`εuq

|t1´t2|ε

¸

ď supt1,t2Pr0,T s,t1ăt2

ˆ

|t1|maxtr`ε´δ,0u

pÁqr`ε´mintδ,r`εu et1ALpHq X0LppP;¨Hmintδ,r`εuq

˙

“ supt1,t2Pr0,T s,t1ăt2

ˆ

|t1|maxtr`ε´δ,0u

pÁqmaxtr`ε´δ,0u et1ALpHq X0LppP;¨Hmintδ,r`εuq

˙

ď X0LppP;¨Hmintδ,r`εuq.

(11.11)


Combining this with the triangle inequality and Proposition 11.3.1 completes theproof of Corollary 11.3.2.


Lemma 11.3.3 (Variances). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|q, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq

, ¨L2pBorelp0,1q;|¨|Rqq,

let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T sˆΩ Ñ H be a mild solution process of the SPDE

dXtpxq “B2

Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ 0 (11.12)

for x P p0, 1q, t P r0, T s. Then it holds thatş1

0E“

|XT pxq|2‰

dx “ř8

n“11´e´2π2n2T

2π2n2 ‰ř8

n“11´e´2π2n2T

π2n2 .

Proof of Lemma 11.3.3. Throughout this proof let A : DpAq Ď H Ñ H be the Lapla-cian with Dirichlet boundary conditions on H. Then observe that for all t P r0, T s itholds that

E

«

›

›

›

›

ż t

0

ept´sqA dWs

›

›

›

›

2

H

ff

“

ż t

0

›

›ept´sqA›

›

2

HSpHqds “

ż t

0

›

›esA›

›

2

HSpHqds

“

8ÿ

n“1

ż t

0

e´2π2n2s ds “8ÿ

n“1

˜

1´ e´2π2n2t

2π2n2

¸

.

(11.13)


11.4 Chapter 8


Lemma 11.4.1. Let r P p1,8q. Thenř8

n“11

n |lnpn`1q|ră 8.

11.4. CHAPTER 8 287


8ÿ

n“1

1

n |lnpn` 1q|r“

1

lnp2q`

1

2 lnp3q`

8ÿ

n“3

1

n |lnpn` 1q|r

ď1

lnp2q`

1

2 lnp3q`

8ÿ

n“3

1

n |lnpnq|rď

1

lnp2q`

1

2 lnp3q`

8ÿ

n“3

ż n

n´1

1

n |lnpnq|rdx

ď1

lnp2q`

1

2 lnp3q`

8ÿ

n“3

ż n

n´1

1

x |lnpxq|rdx

“1

lnp2q`

1

2 lnp3q`

ż 8

2

1

x |lnpxq|rdx “

1

lnp2q`

1

2 lnp3q`

ż 8

lnp2q

1

xrdx

“1

lnp2q`

1

2 lnp3q`

1

pr ´ 1q |lnp2q|pr´1qă 8.

(11.14)


Proposition 11.4.2 (Lower bounds on the convergence speed). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of in-terpolation spaces associated to ´A, let r P R, ρ P r0,8q, let en P L

2pBorelp0,1q; |¨|Rq,n P N, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q it holds that

enpxq “?

2 sinpnπxq, and let v P Hr satisfy

v “8ÿ

k“1

k´12´2r´2ρ

|lnpk ` 1q|´1Rek. (11.15)

Then

(i) it holds that v P Hr`ρzpYεPp0,8qHr`ρ`εq,

(ii) it holds for all ε P p´8, 0s that

supnPN

˜

np2ρ`εq

›

›

›

›

›

v ´nÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

ă 8, (11.16)

and


(iii) it holds for all ε P p0,8q that

supnPN

˜

np2ρ`εq

›

›

›

›

›

v ´nÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

Hr

¸

“ 8. (11.17)

Proof of Proposition 11.4.2. Observe that Lemma 11.4.1 ensures that

8ÿ

k“1

›

›k´12´2r´2ρ

|lnpk ` 1q|´1Rek›

›

2

Hr`ρ

“

8ÿ

k“1

k´1´4r´4ρ|lnpk ` 1q|´2

Rek

2Hr`ρ

“ π4r`4ρ8ÿ

k“1

k´1|lnpk ` 1q|´2

Ră 8.

(11.18)

This implies that v P Hr`ρ. Example 8.1.5 therefore proves Item (ii). In addition,observe that for all ε P p0,8q it holds that

supnPN

¨

˝np4ρ`2εq

›

›

›

›

›

v ´nÿ

k“1

〈ek, v〉H ek

›

›

›

›

›

2

Hr

˛

‚

“ supnPN

˜

np4ρ`2εq8ÿ

k“n`1

〈ek, v〉H ek2Hr

¸

“ supnPN

˜

np4ρ`2εq8ÿ

k“n`1

|〈ek, v〉H |2Rπ4r k4r

¸

“ supnPN

˜

np4ρ`2εq8ÿ

k“n`1

k´1´4r´4ρ|lnpk ` 1q|´2

Rπ4r k4r

¸

“ π4r supnPN

˜

np4ρ`2εq8ÿ

k“n`1

k´1´4ρ|lnpk ` 1q|´2

R

¸

“ 8.

(11.19)

This proves Item (iii). Example 8.1.5 and (11.19) imply that v R pYεPp0,8qHr`ρ`εq.This together with the fact that v P Hr`ρ proves Item (i). The proof of Proposi-tion 11.4.2 is thus completed.


11.4. CHAPTER 8 289

1 function E x e r c i s e P l o t S p a t i a l ( )2 Ngrid = 2 . ˆ ( 0 : 1 9 ) ; Errs = Ngrid ;3 t ic4 for n = 1 : length ( Ngrid )5 Errs (n) = MCErr( Ngrid (n ) , 2ˆ23 , 2ˆ5 ) ;6 end7 toc8 loglog ( Ngrid , Errs ) ;9 hold on

10 loglog ( Ngrid , Ngrid .ˆ( ´1)∗0 .3 , ’ :∗ r ’ ) ;11 loglog ( Ngrid , Ngrid . ˆ ( ´0 .5 )∗0 .3 , ’ : or ’ ) ;12 loglog ( Ngrid , Ngrid . ˆ ( ´0 .25 )∗0 .3 , ’ :∗ r ’ ) ;13 t i t l e ( ’ Error Galerk in approximation ’ ) ;14 xlabel ( ’ S p a t i a l dimension ’ ) ;15 ylabel ( ’ Root mean square e r r o r ’ ) ;16 end1718 function error = SqrdSampleErr (N, top )19 A = ´pi ˆ2∗ ( (N+1): top ) . ˆ 2 ;20 var = (exp(2∗A)´1)./A/2 ;21 gsq = randn (1 , top N) . ˆ 2 ;22 error = sum( var .∗ gsq ) ;23 end2425 function error = MCErr(N, top ,M)26 error = 0 ;27 for m=1:M28 error = error + SqrdSampleErr (N, top ) ;29 end30 error = sqrt ( error/M) ;31 end

Matlab code 11.1: A Matlab code for a solution of Exercise 8.1.15.


100

101

102

103

104

105

106

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Error Galerkin approximation

Spatial dimension

Ro

ot

me

an

sq

ua

re e

rro

r


11.4. CHAPTER 8 291


Lemma 11.4.3 (Elementary estimates for the zeta function). Let s P p1,8q. Then

8ÿ

n“1

n´s ďs

ps´ 1q. (11.20)


8ÿ

n“1

n´s “ 1`8ÿ

n“2

ż n

n´1

n´s du ď 1`8ÿ

n“2

ż n

n´1

u´s du

ď 1`

ż 8

1

u´s du “ 1`1

ps´ 1q“

s

ps´ 1q.

(11.21)


Proposition 11.4.4. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq bethe Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, letT P p0,8q, N P N, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be acylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, let Y : r0, T sˆΩ Ñ H be a naturally-interpolated exponential Euler approximation for the SPDE


with time step size TN, and let X : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-predictablestochastic process which fulfills that for all t P r0, T s it holds P-a.s. that

Xt “

ż t

0


Then it holds for all r P r0, 14q that`

E“

XT ´ YT 2H

‰˘12ď T r

Nr p12´2rq12ď T r

Nr p1´4rq.

Proof of Proposition 11.4.4. Observe that for all r P r0,8q it holds that

E“

XT ´ YT 2H

‰

“ E

«

›

›

›

›

ż T

0

`

epT´sqA ´ epT´tsuT N qA˘

dWs

›

›

›

›

2

H

ff

“

ż T

0

›

›epT´sqA ´ epT´tsuT N qA›

›

2

HSpHqds

“

ż T

0

›

›pÁqr epT´sqA pÁq´r`

eps´tsuT N qA ´ IdH˘›

›

2

HSpHqds

ď

ż T

0

›

›pÁqr epT´sqA›

›

2

HSpHq

›

›pÁq´r`

eps´tsuT N qA ´ IdH˘›

›

2

LpHqds.

(11.24)


Lemma 4.7.7 hence proves that for all r P r0,8q it holds that

E“

XT ´ YT 2H

‰

ď“

TN

‰2rż T

0

›

›p´Aqr epT´sqA›

›

2

HSpHqds “

“

TN

‰2r8ÿ

n“1

ż T

0

n4r π4r e´2π2n2s ds

““

TN

‰2r8ÿ

n“1

1´ e´2π2n2T

2πp2´4rqnp2´4rqď“

TN

‰2r8ÿ

n“1

1

2πp2´4rq np2´4rq

“ πp4r´2q

2

“

TN

‰2r

«

8ÿ

n“1

np4r´2q

ff

.

(11.25)

Lemma 11.4.3 therefore implies that for all r P r0, 14q it holds that

E“

XT ´ YT 2H

‰

ď πp4r´2q

2

“

TN

‰2r p2´ 4rq

p1´ 4rqď

T 2r

p12´ 2rqN2r. (11.26)


11.5 Chapter 9


Proposition 11.5.1. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P

p0,8q, N P N, ϕ : L2pBorelp0,1q; |¨|Rq Ñ R satisfy that for all v P L2pBorelp0,1q; |¨|Rq

it holds that ϕpvq “ v2L2pBorelp0,1q;|¨|Rq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic ba-

sis, let pWtqtPr0,T s be a cylindrical IdL2pBorelp0,1q;|¨|Rq-Wiener process w.r.t. pFtqtPr0,T s,

let Y : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be a naturally-interpolated exponential Eulerapproximation for the SPDE


with time step size TN, and let X : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t P r0, T s it holds P-a.s. that

Xt “

ż t

0


Then it holds for all r P“

0, 12˘

thatˇ

ˇE“

ϕpXT q‰

´ E“

ϕpYT q‰ˇ

ˇ ď T r

Nr π?

2 p1´2rqď

T r

Nr p12´rq.

11.5. CHAPTER 9 293

Proof of Proposition 11.5.1. Throughout this proof let λn P R, n P N, be the realnumbers with the property that for all n P N it holds that λn “ π2n2 and letpH, 〈¨, ¨〉H , ¨Hq be the R-Hilbert space such that

pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, ¨L2pBorelp0,1q;|¨|Rq

, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rqq.

(11.29)Then observe that

ErϕpXT qs “

ż t

0

›

›ept´sqA›

›

2

HSpHqds “

8ÿ

n“1

ż t

0

e´2λnpt´sq ds (11.30)

and

ErϕpYT qs “

ż t

0

›

›ept´tsuT N qA›

›

2

HSpHqds “

8ÿ

n“1

ż t

0

e´2λnpt´tsuT N q ds. (11.31)

Combining (11.30) and (11.31) proves that for all r P“

0, 12˘

it holds that

|ErϕpXT qs ´ErϕpYT qs| “8ÿ

n“1

ż t

0

e´2λnpt´sq`

1´ e´2λnps´tsuT N q˘

ds

ď

8ÿ

n“1

ż t

0

e´2λnpt´sq`

1´ e´2λnps´tsuT N q˘rds

ď

8ÿ

n“1

ż t

0

e´2λnpt´sq`

2λnps´ tsuT Nq˘rds ď

8ÿ

n“1

2rpλnqrT r

N r

ż t

0

e´2λns ds

ď

8ÿ

n“1

T r

2p1´rq pλnqp1´rqN r

“T r πp2r´2q

2p1´rqN r

«

8ÿ

n“1

n´p2´2rq

ff

ďT r πp2r´2q

2p1´rqN r¨p2´ 2rq

p1´ 2rqď

T r

N r p12´ rq

«

supsPr0,12q

p1´ sq

p2π2qp1´sq

ff

ďT r

N r p12´ rq¨p12q

p2π2q12“

T r

π?

2 p1´ 2rqN r.

(11.32)


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