invariant measures for stochastic evolution equations in m ...math.fau.edu/long/jee2010bls.pdf ·...

J. Evol. Equ. 10 (2010), 785–810© 2010 Springer Basel AG1424-3199/10/040785-26, published onlineMay 11, 2010DOI 10.1007/s00028-010-0070-2

Journal of EvolutionEquations

Invariant measures for stochastic evolution equationsin M-type 2 Banach spaces

Zdzislaw Brzezniak, Hongwei Long and Isabel Simao

Abstract. In this paper, we study invariant measures for stochastic evolution equations in M-type 2 Banachspaces. The existence and uniqueness of invariant measure in this general setting are established underglobal (or local) Lipschitz conditions and certain dissipativity conditions.

1. Introduction

The study on invariant measures for (stochastic) dynamical systems is an importanttopic in the theory of (stochastic) dynamical systems. These measures provide certaininvariant characterization such as ergodicity, strong or weak asymptotic stability forthe processes described by the systems. The research on the existence, uniqueness andregularity of invariant measures for stochastic evolution equations (SEE) in Hilbertspaces has received a lot of attention (cf. [18,21,22,32], and references therein). Someuseful methods have been developed in dealing with the invariant measures of SEE’sin Hilbert spaces in terms of different conditions on the coefficients of the SEE’s.For the existence of invariant measures, the Krylov–Bogoliubov criterion is a pow-erful tool. Some efficient methods (so-called compactness method and dissipativitymethod) have been used by Da Prato, Gatarek and Zabczyk [15] and were furtherapplied to study some specific equations. There are mainly three ways to show theuniqueness of invariant measures. The first one is to verify the strong Feller prop-erty (SFP) and irreducibility (I) (cf. [13,14,31,36]). In this case, one usually had toassume that the noise term is nondegenerate. The second one is the so-called Lyapu-nov approach (cf. [26,27,30]). In this case, the SEE’s admit degenerate noise. Thethird one is the dissipative method developed by Da Prato, Gatarek and Zabczyk [15].In this case, the drift coefficients can satisfy some dissipativity (for example, havingcertain polynomial growth). There are also some other methods of proving uniquenessof invariant measures (cf. [32] and some references therein). The regularity of invari-ant measures in the Hilbert space setting has been studied by Da Prato and Zabczyk

Mathematics Subject Classification (2000): 60H15, 35R15Keywords: Existence and uniqueness, Invariant measures, Stochastic evolution equations, M-type 2

Banach spaces.

786 Z. Brzezniak et al. J. Evol. Equ.

[18] via semi-group theory (see Chapter 8 of [18]), and [24] via integration by partsformula (see also some references therein). We should also mention some analyticalmethods in studying invariant measures for infinite dimensional stochastic systemsusing logarithmic Sobolev inequality (cf. [38] and [25]) or Dirichlet form theory (cf.[2–4]).

Now we describe our work in this paper. Brzezniak [5] and Brzezniak and Peszat[10] initiated the study of stochastic evolution equations in Banach spaces, which hassome important applications. We consider the following SEE on an M-type 2 Banachspace E :

{d X (t) = [AX (t)+ F(X (t))]dt + σ(X (t))dW (t), t ≥ 0

X (0) = X0,(1.1)

where (W (t))t≥0 is a cylindrical Wiener process on a separable Hilbert space H and onfiltered probability space A = (�,F , {Ft }t≥0,P), X0 is an E-valued F0-measurablerandom variable, σ : E → R(H, E) (the set of all γ -radonifying operators from Hinto E), A : D(A) → E is the generator of a C0-semi-group S(t) on E , F : E → Ea measurable mapping. In [5], Brzezniak established the existence and uniqueness ofmild solutions to SEE’s on M-type 2 Banach spaces using the theory of stochasticintegrals on M-type 2 Banach spaces, which was developed by Neidhardt [33] in 1978and Dettweiler [19] in 1982 independently. In [10], Brzezniak and Peszat establishedthe space-time continuity of solutions to SPDE’s driven by a homogeneous Wienerprocess. In the next step, we are naturally interested in the invariant measures forthe SEE’s on M-type 2 Banach spaces. Note that Brzezniak and Gatarek [9] studiedthe existence and uniqueness of martingale solutions to the SEE (1.1) as well the exis-tence of an invariant measure in Banach spaces under certain assumptions on A, F andσ . In their paper, the compactness and dissipativity condition on A and F are imposed.In this paper, we are able to show the existence and uniqueness of an invariant measurefor the SEE (1.1) by introducing a different dissipativity condition and getting rid ofthe compactness assumption.

The contents of the paper are organized as follows. In Sect. 2, we prepare somepreliminaries about the structure of the underlying Banach spaces. In Sect. 3, westate and prove our main results on the existence and uniqueness of an invariantmeasure for the SEE (1.1) under global Lipschitz conditions and certain dissipativityconditions. In Sect. 4, we extend the results in Sect. 3 to SEE (1.1) with locally Lips-chitz nonlinearities. Some results on perturbations of semi-groups are provided in the“Appendix A”.

2. Preliminaries

Let (H, 〈·, ·〉) and (E, | · |) be real separable Hilbert space and Banach space,respectively, and let {ek}∞k=1 be a fixed orthonormal basis of H . Here, the separability

Vol. 10 (2010) Invariant measures for SEEs 787

assumptions on H and E are necessary since we are dealing with stochastic processesin these infinite dimensional spaces and the measurability of stochastic processes isessentially depending on the separability of the underlying spaces. Let γ be a can-oncial cylindrical gaussian distribution on H . We say that a bounded linear operatorL : H → E is γ -radonifying if the image γ ◦ L−1 of γ by L has an extension to aσ -additive measure γL on E . The set of all γ -radonifying operators from H into E isdenoted by R(H, E) (cf. [8] and [10]). For L ∈ R(H, E), we put

‖L‖R(H,E) :={∫

E|x |2dγL(x)

} 12

, (2.1)

which, in view of Fernique Theorem, is a finite number. Then, R(H, E) is a separableBanach space under this norm (cf. Lemma 32 in [33]). For a given L ∈ R(H, E) anda bilinear bounded map G : E × E → R, we define

trL G :=∫

EG(x, x)dγL (x). (2.2)

It is easy to prove that

trL G =∞∑

k=1

G(Lek, Lek) =: trH L∗GL , (2.3)

where G : E → E∗ and L∗ : E∗ → H are the operators defined by [G(x)]y :=G(x, y), for x, y ∈ E and 〈L∗x∗, h〉 = x∗(Lh), for h ∈ H, x∗ ∈ E∗.

Let J : E → E∗ denote the duality mapping of E , i.e. for each x ∈ E

J (x) = {l ∈ BE∗ : l(x) = |x |},

where BE∗ = {l ∈ E∗; |l|E∗ ≤ 1}. It is known, see e.g. Corollary 1.5 of Chapter 1 inDeville, Godefroy and Zizler [20], that the norm | · | on E is Gâteaux differentiableon E if and only if J (x) is single-valued for any x ∈ E \ {0}. In this case, the uniqueelement of J (x) is the Gâteaux derivative of |x | at x , i.e. J (x) = (|x |)′.

DEFINITION 2.1. A semi-inner-product on a complex or real vector space V is amapping [·, ·]: V × V → C (or R) such that

(i) [x + y, z] = [x, z] + [y, z].(ii) [λx, y] = λ[x, y], λ ∈ C (or R).

(iii) [x, x] > 0 for x = 0.(iv) |[x, y]|2 ≤ [x, x][y, y] for x, y ∈ V .

Such a vector space V is called a semi-inner-product space.

Any Banach space E can be made into a semi-inner-product space. Accordingto Hahn–Banach theorem, for each x ∈ E , there exists at least one x∗ ∈ E∗ suchthat ‖x∗‖ = ‖x‖ and 〈x, x∗〉 = ‖x‖2. Then, clearly [x, y] = 〈x, y∗〉 for each x, y


∈ E defines a semi-inner-product on E . In this case, we call [·, ·] a compatible semi-inner-product with respect to the norm | · | on E .

From Proposition 4.6 in Long [28], we know that if E is an M-type 2 Banachspace, then E admits an equivalent 2-smooth norm (see Pisier [37]) and a uniquecompatible semi-inner-product [·, ·] on E × E with respect to this equivalent norm.In this case, the semi-inner-product can be defined by

[x, y] := E 〈x, y∗〉E∗ for any x, y ∈ E,

where

y∗ ={

0, if y = 0,|y|J (y), if y ∈ E \ {0}.

For our aim, we need to assume that the norm | · | on E satisfies the following condition

ASSUMPTION 2.2. There exist some p > 1 and a constant K p > 0 such that thefunction f : E � x �→ |x |2p ∈ R is twice Fréchet differentiable on E and

| f ′′(x)|L(E,E;R) ≤ K p|x |2(p−1); x ∈ E . (2.4)

For example, all separable Hilbert spaces and Lr (�,�, ν) for r ≥ 2, (where(�,�, ν) is an arbitrary positive measure space) satisfy the Assumption 2.2 (cf. [20]).From Proposition 2.1 of Long [29], we know that if the Assumption 2.2 is satisfied,then (E, | · |) is an M-type 2 Banach space.

We also impose the following standard assumption on the driving noise:

ASSUMPTION 2.3. We assume that A = (�,F , (Ft )t≥0,P) is a filtered proba-bility space and that (W (t))t≥0 is a H-cylindrical canonical Wiener process definedon A, i.e. for each t ≥ 0, W (t) is a bounded linear operators from H into L2(�,Ft ,P)

such that:

(i) for all t ≥ 0, and ψ, ϕ ∈ H, E [W (t)ψW (t)ϕ] = t〈ψ, ϕ〉H ,(ii) for eachψ ∈ H, W (t)ψ , t ≥ 0, is a real valued (Ft )t≥0-adapted Wiener process.

3. Existence and uniqueness of invariant measure for SEEs with globalLipschitz nonlinearities

We introduce the following assumption on the coefficients of the SEE (1.1).

ASSUMPTION 3.1.

(i) A is the infinitesimal generator of a strongly continuous semi-group (S(t))t≥0

on E.(ii) F is a mapping from E into E and there exists a constant C1 > 0 such that

|F(x)− F(y)| ≤ C1|x − y|, x, y ∈ E .

(iii) σ is a mapping from E into R(H, E) and there exists a constant C2 > 0 suchthat


‖σ(x)− σ(y)‖R(H,E) ≤ C2|x − y|, x, y ∈ E .

(iv) X0 is an E-valued F0-measurable random variable.

We define N 2(E) = {X : [0,∞)×� → E, X |[0,t]×� is B([0, t])×Ft -measurablesuch that

∫ t0 |X (s)|2ds < ∞ P-a.s. for all t ≥ 0}. We say that an E-valued process

X (t), t ≥ 0, is a mild solution to (1.1) iff X ∈ N 2(E)) and the following integralequation is satisfied,

X (t) = S(t)X0+∫ t

0S(t − s)F(X (s)) ds+

∫ t

0S(t − s)σ (X (s))dW (s), a.s. t ≥0.

(3.1)

REMARK 3.2. From Neidhardt [33] (see also [8]), we know that the stochasticintegral

∫ t0 (s)dW (s) is well defined for any R(H, E)-valued adapted process

such that

P

(∫ t

0‖(s)‖2

R(H,E)ds < ∞, t ≥ 0

)= 1.

Moreover,

E

∣∣∣∣∫ t

0(s)dW (s)

∣∣∣∣2

E≤ CE

∫ t

0‖(s)‖2

R(H,E)ds, t ≥ 0

if the right-hand side is finite, where C is a positive constant. Thus, in our SEE setting,the stochastic convolution integral

∫ t0 S(t −s)σ (X (s))dW (s) is well defined provided

that σ(X (s)) is a Ft -adapted R(H, E)-valued process and

∫ t

0‖S(t − s)σ (X (s))‖2

R(H,E)ds < ∞ P − a.s., t ≥ 0.

Before dealing with the existence and uniqueness of a mild solution to the SEE(1.1), we first discuss the continuity of sample paths for the stochastic convolution

u(t) =∫ t

0S(t − s)g(s)dW (s), t ≥ 0, (3.2)

where g ∈ Mq(R(H, E)) for some fixed q > 2, i.e. g(t), t ≥ 0 is a progressivelymeasurable R(H, E)-valued process satisfying

E

∫ T

0‖g(t)‖q

R(H,E) dt < ∞, 0 < T < ∞.

The case when A is a generator of an analytic semi-group has been studied in [9] and[6].

The following two lemmas are proved in Da Prato, Kwapién and Zabczyk [16] forHilbert spaces and the proofs carry over to Banach spaces without change.


LEMMA 3.3. If r > 1, 12 > α > 1

r , y ∈ Lr (0, T ; E) and

x(t) =∫ t

0(t − s)α−1S(t − s)y(s) ds, t ∈ [0, T ]

then x ∈ C([0, T ], E).

Letting 1q < α < 1

2 and setting

y(t) = 1

�(1 − α)

∫ t

0(t − s)−αS(t − s)g(s)dW (s), t ≥ 0, (3.3)

then by the factorization formula, one can prove that

u(t) = 1

�(α)

∫ t

0(t − s)α−1S(t − s)y(s)ds t ≥ 0. (3.4)

LEMMA 3.4. Assume that g ∈ Mq(R(H, E)) for some fixed q > 2. Define pro-cesses y and u by formulae (3.3) and (3.2), respectively. Then (i) y ∈ Lq(0, T ; E)a.s., (ii) u ∈ C([0, T ], E) a.s.

We also have the following standard result.

THEOREM 3.5. Under the Assumption 3.1, there exists a unique mild solutionX (t), t ≥ 0, to SEE (1.1). Moreover, the solution has a continuous modification.

Proof. In order to prove our result, we can follow almost the same arguments in theproof of Theorem 7.4 from the monograph [17] by Da Prato and Zabczyk modifiedto the M-type 2 Banach space setting as it was done for example in [6] in the case ofanalytic semi-groups. We omit the details here. �

Next, we shall deal with the existence and uniqueness of an invariant measure forthe SEE (1.1) under some additional conditions. Let (Pt )t≥0 denote the Markoviantransition semi-group associated with the SEE (1.1) and let P1(E) denote the set ofall probability measures defined on (E,B(E)). Let us recall that

(Ps,tϕ)(x) := E [ϕ(X (t, s; x))] , x ∈ E, t ≥ s, ϕ ∈ Cb(E),

where with x ∈ E , X (t, s; x), t ≥ s, s ∈ R+ is the unique solution of the SEE (1.1)on the time interval [s,∞) with initial data X (s) = x . We also put Pt := P0,t . Then,similar to Theorem 9.8 in [17], we can prove the following result.

THEOREM 3.6. Under the Assumption 3.1, for arbitrary x ∈ E, t ≥ s ≥ r ,ϕ ∈ Cb(E)

E [ϕ(X (t, r; x))|Fs] = (Ps,tϕ)(X (s, r; x)).

In particular, Pr,t = Ps,t Pr,s , if t ≥ s ≥ r and Pt+s = Pt Ps, if t, s ≥ 0.


Note that under Assumption 3.1, the solution X (t, s; x) depends continuously onthe initial data X (s) = x in mean squared sense (see Theorem 9.1 of [17] for Hilbertspaces and its proof carries over to M-type 2 Banach spaces without change). Hence,{Pt }t≥0 is a Feller semi-group, i.e., Ptϕ ∈ Cb(E) for any ϕ ∈ Cb(E) and t ≥ 0. Let usrecall the definition of invariant measure. A probability measure μ ∈ P1(E) is saidto be invariant (or stationary) with respect to Pt , t ≥ 0 if and only if P∗

t μ = μ foreach t ≥ 0. Our main result of this section is the following theorem.

THEOREM 3.7. Let (E, | · |) be a separable Banach space satisfying Assump-tion 2.2 and let [·, ·] be the unique compatible semi-inner-product on E × E. Assumethat A, F and σ satisfy the Assumption 3.1. Furthermore, we assume that there exista constant ω > 0 and a positive integer n0 such that

[An(x − y)+ F(x)− F(y), x − y] + K p

4p‖σ(x)− σ(y)‖2

R(H,E)

≤ −ω|x − y|2, for all x, y ∈ E, n ≥ n0, (3.5)

where An = n A(nI − A)−1 are the Yosida approximations of A. Then, there existsexactly one invariant measure μ for the SEE (1.1). Moreover, it is strongly mixing andfor any ν ∈ P1(E),

P∗t ν → μ weakly as t → +∞,

and there exists C > 0 such that for any bounded Lipschitz continuous functionϕ : E → R,

|Ptϕ(x)−∫

Eϕ(x) dμ(x)| ≤ C(1 + |x |)e− ωt

2p |ϕ|Lip, t > 0, x ∈ E . (3.6)

Proof. We follow the so-called remote start method introduced by Da Prato, Gatarekand Zabczyk [15], see also Da Prato and Zabczyk [18].

Let us introduce another cylindrical Wiener process V (t), t ≥ 0, independent ofW (t), t ≥ 0. Define

W1(t) ={

W (t), if t ≥ 0,V (−t), if t < 0,

and F1(t) = σ(W1(s), s ≤ t), t ∈ R.For any s ∈ R and x ∈ E , we consider the following regularized equation{

d Xn(t) = [An Xn(t)+ F(Xn(t))]dt + σ(Xn(t))dW1(t), t ≥ s,Xn(s) = x ∈ E,

(3.7)

which has a strong solution Xn(t) = Xn(t, s, x), t ≥ s, see Theorem 2.18 of Brzezniakand Elworthy [8] or Theorem 118 of Neidhardt [33]. Furthermore, for each q ≥ 1,there exists a constant Cq,n,t such that

E

(sup

u∈[s,t]|Xn(u, s, x)|q

)≤ Cq,n,t . (3.8)


From Theorem B.1 of Brzezniak and Peszat [10], we know that for q > 2 and allt ≥ s,

limn→∞ E

{∫ t

s|Xn(r, s, x)− X (r, s, x)|qdr

}= 0, (3.9)

and using a proof similar to that of (3.9), we obtain

limn→∞ E{|Xn(t, s, x)− X (t, s, x)|q} = 0, (3.10)

where X (t) = X (t, s, x), t ≥ s is the solution of the equation

{d X (t) = [AX (t)+ F(X (t))]dt + σ(X (t))dW1(t), t ≥ s

X (s) = x ∈ E .(3.11)

We will proceed in two steps.

Step 1-A priori estimate We apply the Ito’s formula (see Theorem A.1 of [10] orTheorem 74 of [33]) to the process Xn(t) and function f (x) = |x |2p, to get

|Xn(t)|2p−|x |2p =∫ t

sE∗〈 f

′(Xn(u)), d Xn(u)〉E + 1

2

∫ t

strσ(Xn(u)) f ′′(Xn(u)) du

=∫ t

s2p|Xn(u)|2(p−1)[An Xn(u)+ F(Xn(u)), Xn(u)] du

+∫ t

s2p|Xn(u)|2(p−1)

E∗〈|Xn(u)|J (Xn(u)), σ (Xn(u))dW1(u)〉E

+1

2

∫ t

strσ(Xn(u)) f ′′(Xn(u)) du. (3.12)

Note that the process∫ t

s 2p|Xn(u)|2(p−1)E∗〈|Xn(u)|J (Xn(u)), σ (Xn(u))dW1(u)〉E

is a real-valued martingale. In fact, it is clear that this process is a local martingale.The martingale property can be easily derived by applying the process to a sequenceof stopping times τK = inf{t ∈ [s,∞) : |Xn(t)| ≥ K }, using Ito’s isometry, (3.8),and letting K → ∞, since limK→∞ τK = ∞ P-a.s. under Assumption 3.1. Thus, bytaking expectations on both sides of (3.12), we have

E(|Xn(t)|2p) = |x |2p+∫ t

sE

{2p|Xn(u)|2(p−1)[An Xn(u)+F(Xn(u)), Xn(u)]

}du

+1

2

∫ t

sE[trσ(Xn(u)) f ′′(Xn(u))] du.

From this, by condition (3.5) with n ≥ n0, we can find some positive constants C3

and C4 such that


d

dt(E[|Xn(t)|2p]) = E

{2p|Xn(t)|2(p−1)[An Xn(t)+ F(Xn(t)), Xn(t)]

+1

2trσ(Xn(t)) f ′′(Xn(t))

}

≤ E

{2p|Xn(t)|2(p−1)[An Xn(t)+ F(Xn(t)), Xn(t)]

+ K p

2|Xn(t)|2(p−1)‖σ(Xn(t))‖2

R(H,E)

}

≤ E

{|Xn(t)|2(p−1)

(2p[An Xn(t)+ F(Xn(t))− F(0), Xn(t)]

+ K p

2‖σ(Xn(t))− σ(0)‖2

R(H,E)

)}

+ 2p E

(|Xn(t)|2(p−1)[F(0), Xn(t)]

)

+E

(K p‖σ(0)‖R(H,E)|Xn(t)|2(p−1)‖σ(Xn(t))− σ(0)‖R(H,E)

)

+ K p

2E

(|Xn(t)|2(p−1)‖σ(0)‖2

R(H,E)

)

≤ −2pωE[|Xn(t)|2p] + E[(2pC3 + K pC4)|Xn(t)|2p−1

+1

2K pC4|Xn(t)|2(p−1)].

Using the Young inequality

xα yβ ≤ αx + βy, x, y, α, β ≥ 0 and α + β = 1,

we get

d

dt(E[|Xn(t)|2p]) ≤ −ωE[|Xn(t)|2p] + C5, n ≥ n0, (3.13)

where C5 is a suitable positive constant depending only on C3,C4, p, K p and ω.Therefore, there exists a constant C6 > 0 such that

E|Xn(t)|2p ≤ e−ω(t−s)|x |2p + C6, t ≥ s, n ≥ n0. (3.14)

Using (3.10) and letting n → ∞ in (3.14), we get

E|X (t)|2p ≤ e−ω(t−s)|x |2p + C6 ≤ C7(1 + |x |2p), t ≥ s, (3.15)

for some constant C7 > 0. It easily follows that there exists a constant C8 > 0 suchthat

E|X (t, s; x)|2 ≤ C8(1 + |x |2), t ≥ s, x ∈ E . (3.16)

Step 2 For δ > λ > 0, we define

Z(t) = X (t,−λ, x)− X (t,−δ, x), t ≥ −λ.


By applying Ito’s formula and proceeding as before, we get the following estimate

E(|Z(t)|2p) ≤ e−ω(t+λ)E|x − X (−λ,−δ, x)|2p. (3.17)

Let t = 0, it follows that

E|X (0,−λ, x)− X (0,−δ, x)|2p ≤ C7(1 + |x |2p)e−ωλ, δ > λ. (3.18)

It immediately yields that

E|X (0,−λ, x)− X (0,−δ, x)|2 ≤ C8(1 + |x |2)e−ωλ/p, δ > λ. (3.19)

This implies that the sequence of random variables {X (0,−λ, x)}λ≥0 is Cauchy inL2(�; E), and consequently as λ → +∞, it converges to a random variable η ∈L2(�; E) which is unique and independent of x . We claim that the law of η is theunique invariant measure with the required properties. To see this it is enough to notethat,

Pt (x, ·) = L(X (t, 0, x)) = L(X (0,−t, x)) → L(η) = μ, weakly as t → +∞,

(3.20)

which is equivalent to

P∗t δx → μ weakly as t → +∞. (3.21)

From (3.21), it follows that for t > 0 and ϕ ∈ Cb(E)

〈ϕ, P∗t μ〉 = 〈Ptϕ,μ〉 = 〈Ptϕ, lim

r→∞ P∗r δx 〉

= limr→∞〈Ptϕ, P∗

r δx 〉 = limr→∞〈Pr+tϕ, δx 〉

= limr→∞〈ϕ, P∗

r+tδx 〉 = 〈ϕ, limr→∞ P∗

r+tδx 〉= 〈ϕ,μ〉,

which implies that P∗t μ = μ for arbitrary t ≥ 0, i.e., μ is an invariant measure for

Pt , t ≥ 0. By (3.20) and Corollary 3.4.3 of Da Prato and Zabczyk [18], we concludethat μ is strongly mixing. On the other hand, (3.20) is equivalent to

limt→∞ Ptϕ = 〈ϕ,μ〉, for all ϕ ∈ Cb(E).

Therefore, if ν ∈ P1(E), we have for all ϕ ∈ Cb(E)

〈ϕ, P∗t ν〉 = 〈Ptϕ, ν〉 → 〈ϕ,μ〉 as t → +∞, (3.22)

by dominated convergence theorem. This implies that

P∗t ν → μ weakly as t → +∞.


We claim that the invariant measure μ is unique. Indeed, if ν is also an invariantmeasure for Pt , we have for all ϕ ∈ Cb(E)

〈ϕ, ν〉 = 〈ϕ, P∗t ν〉 → 〈ϕ,μ〉, as t → ∞,

by (3.22). Therefore, 〈ϕ, ν〉 = 〈ϕ,μ〉 for all ϕ ∈ Cb(E) and we conclude that ν = μ.Finally, let ϕ ∈ Lip(E), then from (3.19) we have

|Ptϕ(x)− Psϕ(x)|2 = |E[ϕ(X (0,−t, x))] − E[ϕ(X (0,−s, x))]|2≤ |ϕ|2LipE|X (0,−t, x)− X (0,−s, x)|2

≤ C8(1 + |x |2)|ϕ|2Lipe− ωp t. (3.23)

Letting s → +∞, we get

|Ptϕ(x)− 〈ϕ,μ〉| ≤ √C8(1 + |x |)|ϕ|Lipe− ω

2p t.

This completes the proof. �

REMARK 3.8. If A is the generator of a strongly continuous semi-group (S(t))t≥0

on E satisfying

‖S(t)‖ ≤ eω1t , t ≥ 0

for some constant ω1 ∈ R, then, by Hille–Yosida theorem, it is easy to show that

[An x, x] ≤ nω1

n − ω1|x |2, x ∈ E, n > ω1.

Therefore, if in addition, there exists a constant ω2 ∈ R with ω1 + ω2 < 0 such that

[F(x)− F(y), x − y] + K p

4p‖σ(x)− σ(y)‖2

R(H,E) ≤ ω2|x − y|2, x, y ∈ E,

then condition (3.5) holds for any ω ∈ (0,−(ω1 + ω2)) and n ≥ ω1(ω+ω2)ω1+ω2+ω .

EXAMPLE 3.9. Let k ∈ L p(R; H) for all p ∈ [2,∞). For each h ∈ H , define

(σh)(ξ) = 〈k(ξ), h〉H .

Then, see Proposition 3.2 of Brzezniak and Peszat [10], for all p ∈ (1,∞), σ ∈R(H, L p(R)).

Let, with p ∈ (1,∞), Ap be the operator

{D(Ap) =

{x ∈ L p(R) ∩ H2,p

loc (R) : ax ′′ + bx ′ + cx ∈ L p(R)}

Apx(ξ) = a(ξ)x ′′(ξ)+ b(ξ)x ′(ξ)+ c(ξ)x(ξ), x ∈ D(Ap), ξ ∈ R,(3.24)


where a(ξ) = 12 a2ξ

2 + a0, b(ξ) = b1ξ and c(ξ) = c0, ξ ∈ R with parametersa2, a0, b1, c0 ∈ R such that 0 < b1 <

14 a2, a0 > 0 c0 < 0 and 1

2 (14 a2 − b1) + c0 −

12 a0 > 0. We should point out here that such parameters do exist.

Note that a function x0 defined by x0(ξ) = 1√π

e− ξ2

2 , ξ ∈ R belongs to L2(R) and

|x0|2L2(R)= 1/

√π . Since x ′

0(ξ) = −ξ x0(ξ) and x ′′0 (ξ) = (ξ2 − 1)x0(ξ) we see that

ax ′′0 , bx ′

0, cx0 ∈ L2(R) so that x0 ∈ D(A2) and

A2x0(ξ) =[

1

2a2ξ

4 − (a2

2+ b1 − a0)ξ

2 + c0 − a0

]x0(ξ), ξ ∈ R.

Thus,

〈A2x0, x0〉L2(R)

=∫

R

[a2

2ξ4 −

(a2

2+ b1 − a0

)ξ2 + c0 − a0

]|x0(ξ)|2 dξ

= 1

π

∫R

[a2

2ξ4 −

(a2

2+ b1 − a0

)ξ2 + c0 − a0

]e−ξ2

dξ

= 1

π

[a2

2

∫R

ξ4e−ξ2dξ −

(a2

2+ b1 − a0

)

×∫

R

ξ2e−ξ2dξ + (c0 − a0)

∫R

e−ξ2dξ

]

= 1√π

[a2

2

3

4−

(a2

2+ b1 − a0

) 1

2+ (c0 − a0)

]

= 1√π

[1

8a2 − 1

2b1 + c0 − 1

2a0

].

By the choice of the constants a2, a0, b1, c0 we infer that 〈A2x0, x0〉L2(R) > 0 andthus A2 is not dissipative. On the other hand, a > 0 and c0 < 0, there exists p > 2such that

c(ξ)+ 1

p

[a′′(ξ)− b′(ξ)

] = c0 + 1

p[a2 − b1] < 0.

Thus, by Theorem 3.2.7 in Fattorini [23], we infer that Ap is m-dissipative in L p(R).Letψ : R → R be a decreasing function, such thatψ(0) = 0 and |ψ(ξ2)−ψ(ξ1)| ≤

K |ξ2 − ξ1|, for all ξ1, ξ2 ∈ R, where K is a constant satisfying 0 < K < a28 − b1

2 +c0 − 1

2 a0.Let F : L p(R) → L p(R) be the Nemytski operator associated with the function

ψ , i.e.

[F(x)](ξ) = ψ(x(ξ)), x ∈ L p(R), ξ ∈ R.

The mapping A2 + F is not dissipative in L2(R). Indeed, it can easily be checked that

〈A2x0, x0〉L2(R) + 〈F(x0), x0〉L2(R) > 0.


On the other hand, if we take p > 2 such that

c0

2+ 1

p(a2 − b1) < 0,

then, the operator Ap − c02 I is m-dissipative in L p(R), i.e. [Apx, x] ≤ c0

2 ‖x‖2p for

x ∈ L p(R) (cf. Theorem 3.2.7 of [23]). We also need to verify that Ap + F is dissi-pative. Using the definition of F and Lipschitz condition on ψ , we find

[F(x), x] =∫ψ(x(ξ))‖x‖2−p

p |x(ξ)|p−2x(ξ)dξ

≤ ‖x‖2−pp

∫K |x(ξ)|pdξ = K‖x‖2

p.

Thus, combining the previous results, we obtain

[Apx, x] + [F(x), x] ≤(c0

2+ K

)‖x‖2

p,

provided that c02 +K < 0, which is satisfied if we impose that a2

8 − b12 +c0− 1

2 a0 < − c02 .

Therefore, in view of the Remark 3.8, the Assumption 3.1 and condition (3.5) in The-orem 3.7 are satisfied by the following SEE on E = L p(R)

d X (t) = [Ap X (t)+ F(X (t))]dt + σdWt .

Compared with the assumptions in Brzezniak and Gatarek [9], we are able to get ridof the compactness condition on the semi-group (or the resolvent of the generator) inTheorem 3.7. Here, in this example, we strongly believe that the semi-group generatedby Ap is not compact so that our example cannot be covered by assumptions in all theprevious papers. Now let us give some convincing arguments to support our claim.

Consider an operator{

D( A) ={

x ∈ L2(R) ∩ H2,2loc (R) : (ax ′)′ − x ∈ L2(R)

}Ax(ξ) = (a(ξ)x ′(ξ))′ − x(ξ), x ∈ D( A), ξ ∈ R,

(3.25)

where a(ξ) = ξ2 +1. Then the operator − A is self-adjoint and positive on L2(R) andmoreover

〈− Ax, x〉 =∫ [

(1 + ξ2)|x ′(ξ)|2 + |x(ξ)|2]

dξ, x ∈ D( A).

It follows that

D( A12 ) =

{x ∈ L2(R) ∩ H1,2

loc (R) :∫ [

(1 + ξ2)|x ′(ξ)|2 + |x(ξ)|2]

dξ < ∞}

and

‖x‖2

D( A12 )

:=∫ [

(1 + ξ2)|x ′(ξ)|2 + |x(ξ)|2]

dξ, x ∈ D( A12 ). (3.26)


We claim that the resolvent of A is not compact and consequently the semi-group gen-erated by A is not compact. In order to prove that the resolvent of A is not compact, it

is enough to show that the natural embedding from D( A12 ) to L2(R) is not compact

(see the proof of Theorem 2.6 in [9]). For this, it is enough to construct a sequence

(xn) of elements from D( A12 ) such that ‖xn‖2

D( A12 )

≤ 100 for all n ∈ N and (xn) is

divergent in L2(R).Motivated by Example 6.11 in Chapter 6 of Adams and Fournier [1], such a sequence

can be constructed as follows. First, we put an = 2n and hn =√

32n . Then, we

define a function vn by requesting that supp(vn) = [0, 2an], vn is symmetric aroundan , vn(an) = hn and vn is linear on [0, an]. By the choice of the numbers an andhn , we infer that ‖vn‖2

L2(R)= 2

3 anh2n = 2 and (a1 + · · · + an)

2∫ |v′

n(ξ)|2 dξ ≤2(a1 + · · · + an)

2 h2n

an= 24(2n − 1)2( 1

2n )2 ≤ 24. We put finally x1 = v1 and xn =

vn(· − 2(a1 + · · · + an−1)). Then, we have ‖xn‖2L2(R)

= ‖vn‖2L2(R)

= 2 and

∫(1 + ξ2)|x ′

n(ξ)|2 dξ ≤ [1 + 4(a1 + · · · + an)2]

∫|v′

n(ξ)|2 dξ

=∫

|v′n(ξ)|2 dξ + 4(a1 + · · · + an)

2∫

|v′n(ξ)|2 dξ

≤ 2anh2

n

a2n

+ 96 ≤ 98.

Hence we proved as claimed that ‖xn‖2

D( A12 )

≤ 100.

On the other hand, the supports of functions xn are pairwise disjoint and ‖xn‖2L2(R)

=2. Hence, there is no subsequence of the sequence (xn)which is convergent in L2(R).

This shows that the resolvent of A is not compact. We will show below by invokingTheorem A.6 that the resolvent of A2 is neither compact. For this let us first observethat the operator B := A2 − A satisfies

Bx = (a′ − b)x ′ − (1 + c0)x =: B1x + B0x, x ∈ D( A).

Since B1x = (a2 −b1)ξ x ′(ξ) and the operator B0 is bounded, it is enough to consideran operator B1 defined by

(B1x)(ξ) = ξ x ′(ξ), x ∈ D( A), ξ ∈ R.

It follows from the above and identity (3.26) that B1 A−1/2 is a bounded operator.Hence, see for instance the proof of Corollary 3.2.4 in [35], that B and A satisfy thecondition (A.4) for any a > 0 in the “Appendix A”. Hence, by invoking Theorem A.6,we infer that the resolvent of A2 = A + B is not compact.

This argument strongly indicates that the semi-groups generated by the operatorsAp are also not compact. However, this question will be investigated in detail in afuture work by the authors.


4. SEEs with locally Lipschitz nonlinearities

We can prove results similar to those in the Sect. 3 under weaker conditions, namelythe local Lipschitz condition for F . We shall first prove that there exists a unique globalmild solution to the SEE (1.1) under local Lipschitz conditions and a dissipativitycondition.

Let us fix constants M > 0 and β ∈ R such that |S(t)| ≤ Meβt , t ≥ 0.

THEOREM 4.1. Let (E, | · |) be a separable Banach space satisfying Assump-tion 2.2 and let [·, ·] be the unique compatible semi-inner-product on E × E. Assumethat F and σ satisfy the following local Lipschitz condition. For every n ∈ N, we canfind cn, dn > 0 such that

|F(x)− F(y)| ≤ cn|x − y|, x, y ∈ Bn, (4.1)

‖σ(x)− σ(y)‖R(H,E) ≤ dn|x − y|, x, y ∈ Bn, (4.2)

where Bn = {x ∈ E : |x | ≤ n}. We further assume that we can find D0 ≥ 0, D1 ∈ R

and D2 > 0, such that

[Ax + F(x), x] ≤ D1|x |2 + D0, x ∈ D(A), (4.3)

‖σ(x)‖2R(H,E) ≤ D2(1 + |x |2), x ∈ E . (4.4)

Then, the SEE (1.1) has a unique global mild solution X (t).

We shall prove Theorem 4.1 by means of several lemmata. We begin with defininga sequence of mappings Fn : E → E and σn : E → R(H, E) by

Fn(x) ={

F(x), if |x | ≤ n,F( x

|x | · n), if |x | > n,(4.5)

σn(x) ={

σ(x), if |x | ≤ n,σ ( x

|x | · n), if |x | > n(4.6)

It follows from the proof of Corollary 3 in [7] that both Fn and σn are globally Lipschitzmaps.

Consider the following approximating SEE

d Xn(t) = [AXn(t)+ Fn(Xn(t))]dt + σn(Xn(t))dWt , Xn(0) = X (0). (4.7)

By Theorem 3.5, there exists a unique global mild solution Xn(t) to (4.7), and thissolution has continuous E-valued paths.

The following two lemmas are standard results.

LEMMA 4.2. If m < n, then Xm(t) = Xn(t) for t ≤ Tm, where

Tm = inf{t ≥ 0 : |Xm(t)| ≥ m}.


LEMMA 4.3. The sequence of stopping times {Tn} is increasing and if we define aprocess X (t) = Xn(t), t ≤ Tn, then X (t) is well defined on {t < T∞}, where

T∞ = limn→∞ Tn = sup

n∈N

Tn .

LEMMA 4.4. If the assumptions of Theorem 4.1 are fulfilled, then T∞ = ∞ a.s.

For the proof of Lemma 4.4, we need the following lemma.

LEMMA 4.5. For any α ∈ R, we have

[x, αy] = α[x, y].

Proof. By Definition 2.1 and the fact that E is a M-type 2 Banach space, we knowthat there exists a unique fy ∈ E∗ such that [x, y] = 〈x, fy〉,| fy |E∗ = |y| and〈y, fy〉 = |y|2. Thus for any α ∈ R and y ∈ E , there exists fαy ∈ E∗ such that

[x, αy] = 〈x, fαy〉.

We claim that fαy = α fy . In fact, we have

|α fy | = |α| · | fy | = |α||y| = |αy|

and

〈αy, α fy〉 = |α|2〈y, fy〉 = |α|2|y|2 = |αy|2 =< αy, fαy > .

This completes the proof. �

Proof of Lemma 4.4. We define a family {Xλn } of new processes by

Xλn (t) = λ(λI − A)−1 Xn(t), t ≥ 0, λ > β ∨ 0, n ∈ N.

Since λ(λI − A)−1x converges to x as λ → ∞ for all x ∈ E , Xλn (t) converges toXn(t) almost surely as λ → ∞. By the Hille–Yosida theorem, we know that

|λ(λI − A)−1| ≤ λM

λ− β, λ > β ∨ 0.

Using Lebesgue Dominated Convergence Theorem, it follows that for any r ≥ 1

limλ→∞ E|Xλn (t)− Xn(t)|r = 0.

Since Xn(t) is the mild solution of SEE (4.7), i.e.

Xn(t) = S(t)X (0)+∫ t

0S(t − s)Fn(Xn(s)) ds +

∫ t

0S(t − s)σn(Xn(s))dW (s),


by applying λ(λI − A)−1 to Xn(t) we get

Xλn (t) = S(t)Xλn (0)+∫ t

0S(t − s)Fλn (Xn(s)) ds +

∫ t

0S(t − s)σλn (Xn(s))dW (s).

where

Fλn (x) = λ(λI − A)−1 (Fn(x)) , x ∈ E,

σ λn (x) = λ(λI − A)−1 ◦ σn(x), x ∈ E .

Then, we infer that Xλn (t) satisfies

Xλn (t) = Xλn (0)+∫ t

0[AXλn (s)+ Fλn (Xn(s))]ds +

∫ t

0σλn (Xn(s))dW (s).

Let us put Aλ = λA(λI − A)−1. Then, we have

Xλn (t) = Xλn (0)+∫ t

0[AλXn(s)+ Fλn (Xn(s))]ds +

∫ t

0σλn (Xn(s))dW (s).

By applying Ito’s formula, see Brzezniak and Elworthy [8], to the function f (x) =|x |2p and the process Xλn (t ∧ Tn), we have

|Xλn (t ∧ Tn)|2p − |Xλn (0)|2p

= 2p∫ t

0|Xλn (s)|2p−2 ([AλXn(s), Xλn (s)] + [Fλn (Xn(s)), Xλn (s)]

)1{s≤Tn} ds

+1

2

∫ t

0trσλn (Xn(s)) f ′′(Xλn (s))1{s≤Tn} ds

+2p∫ t

01{s≤Tn}|Xλn (s)|2p−2[σλn (Xn(s))dW (s), Xλn (s)].

By taking expectations, we obtain

E|Xλn (t ∧ Tn)|2p − E|Xλn (0)|2p

= 2pE

∫ t

0|Xλn (s)|2p−2 ([AλXn(s), Xλn (s)] + [Fλn (Xn(s)), Xλn (s)]

)1{s≤Tn}ds

+1

2E

∫ t

0trσλn (Xn(s)) f ′′(Xλn (s))1{s≤Tn}ds.

Note that for λ > β ∨ 0,

trσλn (Xn(t)) f ′′(Xλn (t)) ≤ K p|Xλn (t)|2p−2‖σλn (Xn(t))‖2R(H,E)

≤ K p D2

(λM

λ− β

)2

|Xλn (t)|2p−2(|Xn(t)|2 + 1).


Hence,

E|Xλn (t ∧ Tn)|2p − E|Xλn (0)|2p

≤ K p D2

2

(λM

λ− β

)2

E

∫ t

0|Xλn (s)|2p−2(|Xn(s)|2 + 1)1{s≤Tn}ds

+2pE

∫ t

0|Xλn (s)|2p−2[AλXn(s)+ Fλn (Xn(s)), Xλn (s)]1{s≤Tn}ds. (4.8)

Basic calculations yield that

[AλXn(t)+ Fλn (Xn(t)), Xλn (t)]1{t≤Tn}= [AXλn (t)+ F(Xλn (t))+ Fλn (Xn(t))− F(Xλn (t)), Xλn (t)]1{t≤Tn}= [AXλn (t)+ F(Xλn (t)), Xλn (t)]1{t≤Tn}

+[Fλn (Xn(t))− F(Xλn (t)), Xλn (t)]1{t≤Tn}. (4.9)

For simplicity, we set

ρλn (t) = [Fλn (Xn(t))− F(Xλn (t)), Xλn (t)]1{t≤Tn}, t ≥ 0.

It is easy to see that ρλn (t) converges to [Fn(Xn(t)) − F(Xn(t)), Xn(t)]1{t≤Tn} = 0almost surely as λ → ∞ and is dominated by C

′n|Xn(t)|. Therefore, using (4.8) and

(4.9), it follows that

E|Xλn (t ∧ Tn)|2p

≤ E|Xλn (0)|2p + K p D2

2

(λM

λ− β

)2

×E

∫ t

0|Xλn (s)|2p−2(|Xn(s)|2 + 1)1{s≤Tn} ds

+2pE

∫ t

0|Xλn (s)|2p−2(D1|Xλn (s)|2 + D0 + ρλn (s))1{s≤Tn} ds. (4.10)

Now, letting λ → ∞ and by Lebesgue Dominated Convergence Theorem and Younginequality, we immediately obtain for any given ε > 0

E|Xn(t ∧ Tn)|2p

≤ E|Xn(0)|2p + M2 K p D2

2E

∫ t

0|Xn(s)|2p−2(|Xn(s)|2 + 1)1{s≤Tn} ds

+ 2pE

∫ t

0|Xn(s)|2p−2(D1|Xn(s)|2 + D0)1{s≤Tn} ds

= E|Xn(0)|2p + (M2 K p D2

2+ 2pD1)E

∫ t

0|Xn(s)|2p1{s≤Tn} ds

+(

M2 K p D2

2+ 2pD0

)E

∫ t

0|Xn(s)|2p−21{s≤Tn} ds


≤ E|Xn(0)|2p +∣∣∣∣ M2 K p D2

2+ 2pD1

∣∣∣∣∫ t

0E|Xn(s ∧ Tn)|2p ds

+(

M2 K p D2

2+ 2pD0

)E

∫ t

0

(|Xn(s ∧ Tn)|2pε

) 2p−22p {(1

ε)

2p−22 } 2

2p ds

≤ E|Xn(0)|2p + Cεt + η

∫ t

0E|Xn(s ∧ Tn)|2p ds, (4.11)

where

η =∣∣∣∣ K p D2 M2

2+ 2pD1

∣∣∣∣ + p − 1

pε

(K p D2 M2

2+ 2pD0

).

Hence, by the Gronwall inequality, it follows that

E|Xn(t ∧ Tn)|2p ≤ E|Xn(0)|2p + Cεt + η

∫ t

0(E|Xn(0)|2p + Cεs)e

η(t−s)ds

= eηtE|Xn(0)|2p + Cε

η(eηt − 1). (4.12)

Noting that Xn(0) = X (0), we have

E|Xn(t ∧ Tn)|2p ≤ eηtE|X (0)|2p + Cε

η(eηt − 1) := Ct < ∞, t ≥ 0.

On the other hand, it is easy to see that

E|Xn(t ∧ Tn)|2p ≥∫

{t>Tn}|Xn(t ∧ Tn)|2pdP ≥ n2p

P{t > Tn}

and so

P{Tn < t} ≤ Ct

n2p. (4.13)

By letting n → ∞, we infer that

P{T∞ < t} = 0

This means that T∞ ≥ t for any t ≥ 0, i.e. T∞ = ∞. �

Now we are in a position to prove Theorem 4.1.

Proof of Theorem 4.1. The existence of a solution X (t) follows from Lemmas 4.2to 4.4. The uniqueness of X (t) follows from that of the mild solution of (4.7). Thiscompletes the proof. �

Finally, we have the following result on the existence and uniqueness of an invariantmeasure for (1.1) under the local Lipschitz conditions on F and σ .


THEOREM 4.6. Let (E, | · |) be a separable Banach space satisfying the Assump-tion 2.2 and let [·, ·] be the unique compatible semi-inner-product on E×E. We assumethat the mapping F from E to E and the mapping σ from E to R(H, E) satisfy thelocal Lipschitz conditions (4.1) and (4.2), respectively. We also assume that (4.3) and(4.4) hold.

Furthermore, we assume that there exist a constant ω > 0 and a positive integerk0 such that

[Ak(x − y)+ F(x)− F(y), x − y] + K p

4p‖σ(x)− σ(y)‖2

R(H,E)

≤ −ω|x − y|2, for all x, y ∈ E, k ≥ k0, (4.14)

where Ak = k A(k I − A)−1 are the Yosida approximations of A. Then, there existsexactly one invariant measure μ for the SEE (1.1), it is strongly mixing and for anyν ∈ P1(E), P∗

t ν → μ weakly as t → ∞. Moreover, there exists C > 0 such that forany bounded Lipschitz continuous function ϕ, all t > 0 and all x ∈ E,

|Ptϕ(x)− 〈ϕ,μ〉| ≤ C(1 + |x |)e− ω2p t |ϕ|Lip.

Proof. The proof is essentially the same as the proof of Theorem 3.7. We shall usethe dissipativity condition (4.14) and follow the step 1 in the proof of Theorem 3.7to obtain an a priori estimate like (3.15). By applying Ito’s formula to the functionf (x) = |x |2p and the process X (k)(t) = X (k)(t, s, x), t ≥ s, which is the uniquestrong solution to the following regularized equation

{d X (k)(t) = [Ak X (k)(t)+ F(X (k)(t))]dt + σ(X (k)(t))dW1(t), t ≥ s,X (k)(s) = x ∈ E,

(4.15)

we obtain

|X (k)(t)|2p − |x |2p

=∫ t

sE∗〈 f

′(X (k)(u)), d X (k)(u)〉E + 1

2

∫ t

strσ(X (k)(u)) f ′′(X (k)(u)) du

=∫ t

s2p|X (k)(u)|2(p−1)[Ak X (k)(u)+ F(X (k)(u)), X (k)(u)] du

+∫ t

s2p|X (k)(u)|2(p−1)

E∗〈|X (k)(u)|J (X (k)(u)), σ (X (k)(u))dW1(u)〉E

+1

2

∫ t

strσ(X (k)(u)) f ′′(X (k)(u)) du. (4.16)

Note that the process

M (k)t =

∫ t

s2p|X (k)(u)|2(p−1)

E∗〈|X (k)(u)|J (X (k)(u)), σ (X (k)(u))dW1(u)〉E

is a real-valued martingale. To verify this, we define S(k)n = inf{t ≥ s : |X (k)(t)| ≥ n}.We can see that S(k)n → ∞ almost surely as n → ∞, as shown in the proof of Lemma


4.4. Applying Ito’s formula to the function g(x) = f (x)2 = |x |4p and the processX (k)(t ∧ S(k)n ), proceeding as in the proof of Lemma 4.4, and using conditions (4.4)and (4.14), we can easily prove that

E|X (k)(t ∧ S(k)n )|4p ≤ C1(t) < ∞, t ≥ s, k ≥ k0, (4.17)

where ∫ t

sC1(r)dr < ∞, t ≥ s.

This implies that M (k)

t∧S(k)nis uniformly integrable for each t ≥ s and therefore the local

martingale M (k)t is a martingale. Thus, by taking expectations on both sides of (4.16)

and proceeding as in the proof of Theorem 3.7, we obtain

E|X (k)(t)|2p ≤ e−ω(t−s)|x |2p + C, t ≥ s, k ≥ k0. (4.18)

We shall now prove that X (k)(t) → X (t) in probability as k → ∞, where X (t) =X (t, s, x), t ≥ s is the global mild solution of the equation

{d X (t) = [AX (t)+ F(X (t))]dt + σ(X (t))dW1(t), t ≥ s

X (s) = x ∈ E .(4.19)

We consider the following approximating SEE{d X (k)n (t) = [Ak X (k)n (t)+ Fn(X

(k)n (t))]dt + σn(X

(k)n (t))dW1(t), t ≥ s,

X (k)n (s) = x ∈ E .(4.20)

Let T (k)n = inf{t ≥ s : |X (k)n (t)| ≥ n} and let Tn be as in Lemma 4.2. It is clearthat X (k)n (t) = X (k)(t) on {t ≤ T (k)n } and Xn(t) = X (t) on {t ≤ Tn}. We putτ(k)n = T (k)n ∧ Tn . Then, by Markov inequality, we have for all ε > 0

P{|X (k)(t)− X (t)| > ε}≤ P[{|X (k)(t)− X (t)| > ε} ∩ {t ≤ τ (k)n }] + P{t > τ(k)n }≤ P[{|X (k)n (t)− Xn(t)| > ε} ∩ {t ≤ τ (k)n }] + P{t > τ(k)n }≤ P{|X (k)n (t)− Xn(t)| > ε} + P{t > τ(k)n }≤ ε−2p

E[|X (k)n (t)− Xn(t)|2p] + P{t > T (k)n } + P{t > Tn}. (4.21)

Similar to the proof of Lemma 4.4, we can show that

P(t > T (k)n ) ≤ C(t)

n2p, k ≥ k0

where C(t) does not depend on k. From this and (4.13), it follows that, for arbitraryδ > 0, we can choose n = n0 large enough so that

P{t > T (k)n0} < δ

3, k ≥ k0, and P{t > Tn0} <

δ

3. (4.22)


For such a fixed n = n0, all the coefficients of (4.7) and (4.20) satisfy global Lipschitzconditions and consequently using (3.10) we can choose a positive integer k1 > k0

large enough such that

E[|X (k)n0(t)− Xn0(t)|2p] ≤ ε2pδ

3, k ≥ k1. (4.23)

Then, combining (4.21), (4.22), and (4.23), we conclude that X (k)(t) → X (t) in prob-ability as k → ∞. Therefore, letting k → ∞ in (4.18) and using Fatou Lemma, weget

E|X (t)|2p ≤ e−ω(t−s)|x |2p + C, t ≥ s, (4.24)

which is the desired a priori estimate.The remaining part of the proof is the same as that of the proof of Theorem 3.7 and

we omit the details here. �

Acknowledgments

This paper was finished during the visit of the first named author to the Depart-ment of Mathematical Sciences, Florida Atlantic University. He would like to thankthe Department for hospitality. Brzezniak has been partially supported by an EP-SRC grant EP/G019584/1. Long is partially supported by FAU start-up funding at theC. E. Schmidt College of Science. Simão is partially supported by FCT, FinanciamentoBase 2008—ISFL/1/209 and POCTI/MAT/34711/99. The authors are grateful to ananonymous referee for the insightful comments and suggestions which have improvedthe presentation of the paper.

Appendix A. Some results on perturbations of semi-groups

We begin with recalling Theorem 3.2.1 from [35].

THEOREM A.1. Let A be an infinitesimal generator of an analytic semi-group ona Banach space X. Then there exists a number δ > 0 such that for every closed linearoperator B in X satisfying D(B) ⊃ D(A) and, for some b ≥ 0,

|Bx | ≤ δ|Ax | + b|x |, x ∈ D(A), (A.1)

A + B is an infinitesimal generator of an analytic semi-group on X.

DEFINITION A.2. The supremum of the set of all δ > 0 for which the conclusionof Theorem A.1 holds true will be denoted by δ0(A).


REMARK A.3. It follows by analysing and slightly modifying the proof of Theo-rem A.1 that

δ0(A) ≥ 1

1 + M,

where M is such that for some ω > 0 and R > 0,

|(λI − A)−1| ≤ M |λ|−1, λ ∈ �(ω, R), (A.2)

where �(ω, R) = {λ ∈ C : |λ| ≥ R, | arg λ| ≤ π/2 + ω}.Moreover, by further analysis and modification, one can prove that if B satisfies (A.1)with δ < 1

2+M , then

δ0(A + B) ≥ 1

1 + M.

The next result strengthens the previous one by considering compact semi-groups.

THEOREM A.4. In the framework of the above Theorem A.1, if A has a compactresolvent, A + B has a compact resolvent.

Proof. The following identity is established in the proof of Theorem A.1

(λI − (A + B))−1 = (λI − A)−1(

I − B(λI − A)−1)−1

(A.3)

for |λ| big enough. It follows from A.3 that if the operator (λI − A)−1 is compact, sois (λI − (A + B))−1. �

Now we formulate the following auxiliary result.

PROPOSITION A.5. Let A be an infinitesimal generator of an analytic semi-groupon a Banach space X and let B be closed linear operator in X satisfying D(B) ⊃ D(A)and, for some a ∈ (0, 1) and b ≥ 0,

|Bx | ≤ a|Ax | + b|x |, x ∈ D(A). (A.4)

Then,

|Bx | ≤ a

1 − a|(A + B)x | + b

1 − a|x |, x ∈ D(A). (A.5)

Proof. By the triangle inequality we have

|(A + B)x | ≥ |Ax | − |Bx |, x ∈ D(A).

Hence, by taking into account (A.4), we infer that

|(A + B)x | ≥ |Ax | − a|Ax | − b|x | = (1 − a)|Ax | − b|x |, x ∈ D(A).


Therefore, since a < 1,

|Ax | ≤ 1

1 − a|(A + B)x | + b

1 − a|x |, x ∈ D(A).

Finally, by applying (A.4) to the last inequality we conclude the proof. �

We conclude this section with the following result.

THEOREM A.6. Let A be an infinitesimal generator of an analytic semi-group ona Banach space X and let M be a constant such that the condition (A.2) is satisfied.Let us also assume that B is closed linear operator in X satisfying D(B) ⊃ D(A)and (A.4), for some b ≥ 0 and

a ∈ (0, 1

2 + M). (A.6)

Then, if A has a noncompact resolvent, then A + B has a noncompact resolvent aswell.

Proof. Follows by arguing by contradiction from Theorem A.4, Proposition A.5 andRemark A.3. Note that the condition (A.6) implies that a

1−a <1

1+M . �

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Z. BrzezniakDepartment of MathematicsUniversity of York,Heslington, York YO10 5DD, UKE-mail: [email protected]

H. LongDepartment of Mathematical SciencesFlorida Atlantic University,Boca Raton, FL 33431, USAE-mail: [email protected]

I. SimãoCMAF-Universidade de LisboaAv. Prof. Gama Pinto 21649-003 LisboaPortugalE-mail: [email protected]

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