to the chairman of examiners for part iii mathematics.math.mit.edu/~sswatson/pdfs/partiiiessay.pdfi...

To the Chairman of Examiners for Part III Mathematics.

Dear Sir,

I enclose the Part III essay of Sam Watson.

Signed (Director of Studies)

Brownian motion, complex analysis, and the dimension of theBrownian frontier

Sam WatsonTrinity College

Cambridge University

30 April, 2010

I declare that this essay is work done as part of the Part III Examination.I have read and understood the Statement on Plagiarism for Part III andGraduate Courses issued by the Faculty of Mathematics, and have abidedby it. This essay is the result of my own work, and except where explicitlystated otherwise, only includes material undertaken since the publicationof the list of essay titles, and includes nothing which was performed incollaboration. No part of this essay has been submitted, or is concurrentlybeing submitted, for any degree, diploma or similar qualification at anyuniversity or similar institution.

Signed:

58 County Road 362Oxford, Mississippi 38655

USA

Brownian motion, complex analysis, and the dimension of theBrownian frontier

Contents

1 Introduction 1

2 Brownian Motion and Complex Analysis 12.1 One-dimensional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Brownian motion in Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Conformal invariance of planar Brownian motion . . . . . . . . . . . . . . . . . . . . 72.4 Applications to complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Applications to planar Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Schramm-Loewner evolutions and the dimension of the Brownian frontier 113.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Brownian excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Reflected Brownian excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Loewner’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Restriction property of SLE8/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Hausdorff dimension of SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 Hausdorff dimension of the Brownian frontier . . . . . . . . . . . . . . . . . . . . . . . 21

iv

1 INTRODUCTION

This essay explores the interplay between complex analysis and planar Brownian motion. In Sec-tion 2, we develop basic properties of Brownian motion and give probabilistic proofs of classicaltheorems from complex analysis. We discuss conformal invariance and illustrate how the theoryof conformal maps in can be used to prove statements about planar Brownian motion. Section 3culminates in a proof that the frontier of planar Brownian motion has Hausdorff dimension 4/3almost surely. This proof is made possible by a recent development called the Schramm-Loewnerevolution, which makes extensive use of complex analysis to study random processes in the plane.In preparation for the proof, we develop requisite material on Brownian excursions, Schramm-Loewner evolutions, and Hausdorff dimension.

The material presented in the sections on Brownian motion and its applications to complex analy-sis drawn mainly from Rogers and Williams’s book [11] and Nathanaël Berestycki’s notes on Stochas-tic Calculus [2]. For the review of Schramm-Loewner evolutions and Brownian excursions, JamesNorris’s lecture notes [8] and Gregory Lawler’s book [5] are used. Theorem 3.3.3 on reflected Brow-nian excursions comes from Lawler, Schramm, and Werner’s paper on conformal restriction mea-sures [7]. The heuristic proof given for the upper bound for the dimension of the SLE curves isfrom a survey paper of John Cardy [3], and the proofs given in the final subsection are my own.The idea to use SLE8/3 and reflected Brownian excursions to prove that the dimension of the pla-nar Brownian frontier is 4/3 is due to Vincent Beffara [1].

2 BROWNIAN MOTION AND COMPLEX ANALYSIS

2.1 One-dimensional Brownian motion

Definition 2.1.1. A standard Brownian motion (Bt )t≥0 is a real-valued stochastic process definedon a probability space (Ω,F,P) which satisfies the following conditions.

(i) B0 = 0 almost surely,(ii) for almost all ω ∈Ω, t 7→ Bt (ω) is continuous, and

(iii) for all s, t ≥ 0, Bt+s −Bt is independent of (Bu)0≤u≤t and has distribution N(0,h),

where N(m,σ2) denotes the normal distribution with mean m and variance σ2.

Definition 2.1.2. Given a filtration (Ft )t≥0 on the probability space (Ω,F,P), we say that (Bt )t≥0 isan (Ft )t≥0-Brownian motion if the following conditions hold.

(i) B0 = 0 almost surely,(ii) for almost all ω ∈Ω, t 7→ Bt (ω) is continuous,

(iii) (Bt )t≥0 is adapted to (Ft )t≥0 (i.e., Bt is Ft -measurable for all t ≥ 0), and(iv) for all s, t ≥ 0, Bt+s −Bt is independent of Ft and has distribution N(0,h).

If (Bt − X )t≥0 is a standard Brownian motion for some random variable X , we say that (Bt )t≥0 isa Brownian motion started at X . See [2] for a proof of the following theorem, which confirms theexistence of Brownian motion.

Theorem 2.1.3. (Wiener) There exists a process (Bt )t≥0 satisfying conditions (i)-(iii) in Definition2.1.1.

Idea of proof. Define Dn = 0, 1

2n , 22n , 3

2n , . . . ,1, and let Zn,t : n ∈Z+,d ∈ Dn be a countable collec-

tion of iid standard normal random variables. Set B (0)0 = 0, B (0)

1 = Z0,1, and B (0)t = t Z0,1 (i.e., linearly

2 BROWNIAN MOTION AND COMPLEX ANALYSIS 2

interpolated between the values at 0 and 1). Define B (n−1)t inductively so that B (n)

t = B (n−1)t for

t ∈ Dn−1. For t ∈ Dn \ Dn−1, set t− = t −2−n , t+ = t +2−n , and

B (n+1)t =

B (n−1)t− +B (n−1)

t+

2+ Zn,t

2n

In other words, we have added a Gaussian fluctuation with the appropriate variance at the newdyadic times. Again, extend B (n)

t to [0,1] by linear interpolation. Using some elementary estimates,one can show that B (n)

t is uniformly Cauchy, from which it follows that there exists a uniform limit

Bt . This limit inherits independent, normally distributed intervals from(B (n)

t

)t∈Dn

. We obtain a

Brownian motion on [0,∞) by summing a countable collection of independent copies of Brownianmotion defined on unit intervals.

An adapted process (X t )t≥0 for which (X t1 , X t2 , . . . , X tn ) is jointly Gaussian for all finite sets of times(tk )1≤k≤n is called a Gaussian process. If EX t = 0, then (X t )t≥0 is called a mean-zero process. Itis straightforward to check that a process is a Brownian motion if and only if it is a continuous,mean-zero Gaussian process with covariance EX t Xs = s ∧ t for all s, t ≥ 0.

Definition 2.1.4. The Wiener space W is defined to be the space C ([0,∞),R) of continuous real-valued functions on [0,∞), with the metric

d( f , g ) =∞∑

n=12−n sup

0≤x≤n| f (x)− g (x)|.

We equip W with its Borelσ-algebra W =B(W ). (We use the notation B(X ) for the Borelσ-algebraon a topological space X .)

We may think of a Brownian motion B as a mapΩ→W which sendsω to the continuous functiont 7→ Bt (ω). The image measure of B is called the Wiener measure and is denotedW.

Proposition 2.1.5. Let (Bt )t≥0 be a Brownian motion.

(i) (−Bt )t≥0 is a Brownian motion.(ii) If c > 0, then cBt/c2 is a Brownian motion (scaling).

(iii) The process starting at 0 and equal to tB1/t for t > 0 is a Brownian motion (time inversion).

Proof. The processes (i), (ii), and (iii) are mean-zero Gaussian processes, since they inherit jointlyGaussian finite-dimensional distributions from (Bt )t≥0. To check the covariance, we computeE(−Bs ,−Bt ) = E(Bs ,Bt ) = s ∧ t for (i),

E[(cBs/c2 )(cBt/c2 )

]= c2E[Bs/c2 Bt/c2

]= c2(s/c2 ∧ t/c2) = s ∧ t

for (ii), and

E [(sB1/s)(ctB1/t )] = st E [B1/sB1/t ] = st (1/s ∧1/t ) = min(s, t )

for (iii). The only thing left to prove is continuity at zero for (iii). This follows since (tB1/t )t>0

and (Bt )t>0 are both mean-zero Gaussian processes on C ((0,∞),R) and therefore have the samedistribution as C ((0,∞),R)-valued random variables. Thus

limsupt→0

|Bt | = 0, a.s. ⇒ limsupt→0

|tB1/t | = 0, a.s.

Definition 2.1.6. Let (Ω,F,P, (Ft )t≥0) be a filtered probability space. An R+-valued random vari-able T for which T ≤ t ∈Ft for all t ≥ 0 is called an (Ft )t≥0-stopping time. Given a stopping timeT , the σ-algebra FT is defined by

FT = A ∈F : A∩ T ≤ t ∈Ft for all t ≥ 0.


The following theorem says that a Brownian motion ‘restarted’ at a stopping time T is a new Brow-nian motion independent of the past. For a proof, see [2].

Theorem 2.1.7. (Strong Markov Property of Brownian motion) Let (Bt )t≥0 be a Brownian motion,and let (Ft )t≥0 be its natural filtration. If T is an (Ft )t≥0-stopping time, then the process (BT+t −BT )t≥0 is a Brownian motion with respect to the filtration (FT+t )t≥0 and is independent of FT .

If T is a constant stopping time, then the strong Markov property is sometimes called the Markovproperty or the simple Markov property. We give one consequence of the Markov property whichwe will need later.

Proposition 2.1.8. With probability 1, limsupt Bt =∞ and limsupt Bt =−∞.

Proof. (From [11]). Let X = supt≥0 Bt . By scaling, c X has the same distribution as X for all c > 0.Therefore, X ∈ 0,∞ almost surely. Let p =P(X = 0). Then

p ≤P(X1 ≤ 0 and X1+t ≤ 0∀ t ≥ 0)

=P(X1)P(X1+t ≤ 0∀ t ≥ 0) (by the Markov property)

= 12 p,

which implies p = 0.

Proposition 2.1.9. For all ε > 0, the set Bt : 0 ≤ t ≤ ε almost surely contains both positive andnegative numbers.

Proof. Apply Proposition 2.1.5(iii) to Proposition 2.1.8. Since Bt almost surely takes on arbitrarilylarge and small values as t ranges over the interval (ε−1,∞), we find that (tB1/t )t>0 almost surelytakes on positive and negative values as t ranges over the interval (0,ε).

We record the following important result about martingales. For a proof, see [11]. Recall that amartingale (Mt )t≥0 is defined to be an adapted process for which E|Mt | <∞ for all t ≥ 0 and

E(Mt −Ms |Fs) = 0, ∀0 ≤ s ≤ t <∞.

Also, recall that a collection X of random variables is said to be uniformly integrable if

supX∈X

E(|X |1|X |>K ) → 0 as K →∞.

Theorem 2.1.10. (Optional Stopping Theorem) Let (Mt )t≥0 be a continuous adapted process forwhich E|Mt | <∞ for all t ≥ 0. Then the following are equivalent.

(i) (Mt )t≥0 is a martingale.(ii) (MT∧t )t≥0 is a martingale for all bounded stopping times T .

(iii) E(XT |FS) = XS∧T for all bounded stopping times S,T .(iv) EX0 = EXT for all bounded stopping times T .

Also, if (Mt )t≥0 is a uniformly integrable martingale, then E(XT |FS) = XS∧T for all stopping timeS,T .

The following proposition allows us to give bounds for the probability that sup|Bt | : 0 ≤ t ≤ ε < x.The proof is from [2].

Proposition 2.1.11. Define St = sup0≤s≤t Bs . Then for all t ≥ 0, St has the same distribution as |Bt |.


Proof. For all t ≥ 0 and a ≥ 0, we have

P(St ≥ a) =P(St ≥ a and Bt ≤ a)+P(St ≥ a and Bt > a)

=P(Bt ≥ a)+P(Bt > a)

= 2P(Bt ≥ a) =P(|Bt | ≥ a).

The step P(St ≥ a and Bt ≤ a) = P(Bt ≥ a) comes from the reflection principle [2], which assertsthat for all a,b ∈R, P(St ≥ a and Bt ≤ b) =P(Bt ≥ 2a −b).

We recall a few basic facts from stochastic calculus which are needed for the proof of the followingproposition. Proofs of these statements may be found in [2]. A real-valued process (Mt )t≥0 is saidto be a local martingale if there exists a sequence (Tn)n≥1 of stopping times tending to ∞ as n →∞for which (MTn∧t )t≥0 is a martingale for all n. For every continuous local martingale, there exists aunique continuous adapted nondecreasing process, denoted [M ]t and called the quadratic varia-tion of (M)t≥0, for which (M 2 − [M ]t )t≥0 is a continuous local martingale. A bounded continuouslocal martingale is a true martingale, and the quadratic variation of a Brownian motion (Bt )t≥0 ist .

Proposition 2.1.12. Let 0 < a < b, let B be a Brownian motion started at a, and let τr = inft ≥ 0 :Bt = r . Then P(τ0 > τb) = a/b, and E(τ0 ∧τb) = a(b −a).

Proof. Let p = P(τ0 > τb), and apply the optional stopping theorem to the bounded martingale(Bt )t∧τ0∧τb to find that a = EM0 = EMτ0∧τb = pb + (1 − p)0. For the second equality, note thatB 2

t∧τ0∧τb− t ∧τ0∧τb is a martingale. Therefore, EB 2

t∧τ0∧τb−Et ∧τ0∧τb = EB 2

0 = a2. Take t →∞ andapply dominated convergence for the first term and monotone convergence for the second termto find

b2(a/b)+0−Eτ0 ∧τb = a2,

which gives Eτ0 ∧τb = a(b −a).

2.2 Brownian motion in Rd

Definition 2.2.1. A Brownian motion in Rd is a d-dimensional vector whose components are in-dependent scalar Brownian motions. A planar Brownian motion is a Brownian motion in R2.

Although Brownian motion is defined using Cartesian coordinates, we shall see that it is rotation-ally invariant (Proposition 2.2.3). In preparation, we state the following theorem from stochasticcalculus. For a proof, see [2].

Theorem 2.2.2. (Lévy’s characterisation) Let M = (M 1, . . . , M d ) be an Rd -valued random processwhose components are continuous local martingales. Then M is a Brownian motion if and only if

[M i , M j ]t =

t if i = j0 if i , j .

Proposition 2.2.3. If B is a Brownian motion in Rd and U is an orthogonal matrix (that is, UU T =I ), then U B is a Brownian motion.

Proof. Let u1,u2, . . . ,ud be the rows of the matrix U . As a linear combination of continuous localmartingales, the components of U B are continuous local martingales. Also, the i th componentof U B is (U B)i = uT

i B , so [(U B)i , (U B) j ]t = uTi u j t , which is t if i = j and is 0 otherwise, by the

definition of orthogonality. By Lévy’s charactisation, U B is a Brownian motion.


Given a d-dimensional Brownian motion (Bt )t≥0, it is often useful to show that ( f (Bt ))t≥0 is a mar-tingale for some carefully chosen function f . This technique is especially useful when d ≥ 2, as wewill now demonstrate with a theorem and an example. We follow the presentation in [11].

Theorem 2.2.4. Let f : [0,∞)×Rd → R be continuously differentiable in the first coordinate andtwice continuously differentiable in the second coordinate. Suppose that there exists K > 0 forwhich

| f (t , x)|+∣∣∣∣∂ f

∂t(t , x)

∣∣∣∣+ d∑i=1

∣∣∣∣ ∂ f

∂xi(t , x)

∣∣∣∣+ d∑i , j=1

∣∣∣∣ ∂2 f

∂xi∂x j(t , x)

∣∣∣∣≤ K exp(K (t +|x|), (2.2.1)

for all (t , x) ∈ [0,∞)×Rd . Then

Ct B f (t ,Bt )− f (0,B0)−∫ t

0

(∂

∂t+ 1

24

)f (s,Bs)d s

is a martingale. Here 4 denotes the Laplacian∑d

i=1∂2/∂x2

i . In particular, if f is harmonic (that is,4 f is identically zero), then f (t ,Bt ) is a martingale.

Proof. First we will show that E(Ct−Cs) = 0 for 0 < s ≤ t . Denote by pt (x) = exp(−|x|2/2t )/(2πt )−d/2

the density of Bt . We compute

E(Ct −Cs) = E(

f (t ,Bt )− f (s,Bs)−∫ t

s

(∂

∂t+ 1

24

)f (u,Bu)du

)=

∫Rd

(pt (x) f (t , x)−ps(x) f (s, x)

)d x −

∫ t

s

∫Rd

pu(x)

(∂

∂t+ 1

24

)f (u, x)d x du

=∫Rd

(pt (x) f (t , x)−ps(x) f (s, x)

)d x −

∫ t

s

∫Rd

∂

∂u(pu(x) f (u, x))d x du,

where we have used integration by parts twice to rewrite∫

pu(x)4 f (u, x)d x as∫ 4pu(x) f (u, x)d x.

Here we require the growth condition (2.2.1) so that the product pu(x) f (u, x) decays sufficientlyrapidly as |x|→∞ that the boundary terms in the integration by parts are zero. We have also used124pu(x) = ∂pu(x)/∂u. Now we may interchange the integrals in the second term, since the mod-ulus of the integrand is bounded. An appeal to the fundamental theorem of calculus completesthe proof that E(Ct −Cs) = 0.

Now we will to take s → 0 to show that E(Ct ) = 0. First, observe that Ct −Cs →Ct pointwise as s → 0.By Propositions 2.1.11 and 2.1.5(ii), we have E(sup0≤u≤t |Bu | ≥ a) ≤ 2E(|B1| ≥ a/

pt ). Thus

|Ct −Cs | =∣∣∣∣ f (t ,Bt )− f (s,Bs)−

∫ t

s

(∂

∂u+ 1

24

)f (u,Bu)du

∣∣∣∣≤ | f (t ,Bt )|+ | f (s,Bs)|+

∫ t

s

∣∣∣∣( ∂

∂u+ 1

24

)f (u,Bu)

∣∣∣∣ du

≤ (2+ t )K exp

(K

(t + sup

0≤u≤t|Bu |

)),

which is integrable because

Eexp

(K sup

0≤u≤t|Bu |

)≤ 1+

∞∑n=1

P

(exp

(K sup

0≤u≤t|Bu |

)≥ n

)≤ 1+2

∞∑n=1

P

(|X1| ≥

logn

Kp

t

)≤ 1+4

∞∑n=1

Kp

tp2π logn

exp(−(logn)2/2K 2t ) <∞.


We have used the estimate E(X1 ≥ x) ≤ x−1

2π exp(−x2/2) for the tail of the normal distribution. ThusE(Ct ) = 0 for all t . By the Markov property of Brownian motion, this gives E(Ct −Cs |Fs) = 0, asdesired.

Theorem 2.2.5. Planar Brownian motion is neighbourhood-recurrent but does not visit a pre-specified point. More precisely, let Bt be a planar Brownian motion started at z0 ∈ C. Then forall open U ⊂C, the set

t ≥ 0 : Bt ∈U (2.2.2)

is almost surely unbounded, and for all w , z0,

t ≥ 0 : Bt = w =;, (2.2.3)

almost surely.

Proof. Without loss of generality, suppose z0 , 0. We will show that with probability 1, the Brow-nian motion visits every neighbourhood of 0 infinitely often but does not hit 0. Let 0 < a < b,and multiply the function z 7→ log |z| by a smooth function which is equal to 1 on z : a ≤ |z| ≤ band 0 on z : a/2 ≤ |z| ≤ 2b to obtain a smooth, compactly supported function f which satisfiesf (z) = log |z| for all a < |z| < b. Then f trivially satisfies (2.2.1). Therefore, f (Bt ) is a martingale.

Also, 4 f = ∂2 f∂x2 + ∂2 f

∂y2 = y2−x2

y2+x2 + x2−y2

y2+x2 = 0 on z : a < |z| < b. Applying the optional stopping theorem

to the uniformly integrable martingale f (Bt ) with the stopping time τ= inft ≥ 0 : |Bt | ∈ a,b, wefind that the probability p that the Brownian motion hits z : |z| = a before z : |z| = b solves

log |z0| = p log a + (1−p) logb.

Thus p = logb−log z0logb−log a . Letting b →∞ gives (2.2.2). Also, letting a → 0 gives (2.2.3), since

P(B hits 0) =∞⋃

n=1P(B hits 0 before z : |z| = n)

=∞⋃

n=1

⋂m>1/|z0|

P(B hits z : |z| = 1/m before z : |z| = n).

The next example is an exercise in [2].

Proposition 2.2.6. If d ≥ 3, then the probability that a d-dimensional Brownian motion started atz ∈Rd never visits the closed ball of radius 0 < r < |z| centered at the origin is (r /|z|)d−2.

Proof. Let Bt be a d-dimensional Brownian motion, and let τr denote the hitting time of the closedball of radius r centered at the origin. Since the function z 7→ |z|2−d is harmonic in Rd \0, we findusing Theorem 2.2.4 that Mt B |Bt∧τr |2−d is a martingale (after smoothing the singularity at theorigin as in the proof of Theorem 2.2.2).

Let s > |z|, let τs be the hitting time of z : |z| = s, and let φ(x) = x2−d . The optional stoppingtheorem applied to Mt implies

φ(x) = EM0 = EMτr ∧τs =P(τr < τs)φ(r )+P(τr ≥ τs)φ(s),

which gives P(τr < τs) = φ(s)−φ(|z|)φ(s)−φ(r )

. Taking s →∞, we find P(τr <∞) = (r /|z|)d−2.


2.3 Conformal invariance of planar Brownian motion

A connected open subset ofC is called a domain. Let D be a domain, let f : D →Cbe a nonconstantanalytic function, and let (BT∧t )t≥0 be a Brownian motion started at z ∈ D and stopped at the exittime T = inft ≥ 0 : Bt ∈ C \ D. By the open mapping theorem [9], f (D) is an open set. Therefore,if we consider the image of (BT∧t )t≥0 under f , we obtain a C-valued stochastic process started atf (z) in the domain f (D) and stopped when it hits C\ f (D) (see Figure 1).

f (z) = exp(3z/2)

Figure 1: A Brownian motion started in the unit disk and its image under theanalytic function z 7→ exp(3z/2).

For all z0 ∈ D with f ′(z0), 0, the Taylor series development

f (z) = z0 + f ′(z0)(z − z0)+O(|z − z0|2),

shows that f transforms points near z0 by scaling z − z0 by a factor of | f ′(z0)| and rotating z − z0

by an angle arg f ′(z0). Since Brownian motion is invariant under rotations and scaling (the latterwith a time change), we expect that f (Bt )t≥0 can be transformed into a Brownian motion with asuitable time change. (The restriction that f ′(z0) , 0 does not pose a problem since the zeros off are isolated). This property of Brownian motion is called conformal invariance, in reference tothe case where f is a conformal map (that is, an injective analytic function). In preparation for aproof, we recall two theorems from stochastic calculus. Proofs may be found in [2].

Theorem 2.3.1. (Itô’s formula) Let D be a domain in Rd+1, let f be a twice continuously differen-tible function from D to R, and let M 1, . . . , M d be continuous local martingales. Then up to the exittime of (t , M 1

t , . . . , M dt ) from D , we have

f (t , M 1t , . . . , M d

t ) = f (0, M 10 , . . . , M d

0 )+∫ t

0

∂ f

∂t(s, M 1

s , . . . , M ds )d s

+d∑

i=1

∫ t

0

∂ f

∂xi(s, M 1

s , . . . , M ds )dM i

s

+ 1

2

d∑i=1

d∑j=1

∫ t

0

∂2 f

∂xi∂x j(s, M 1

s , . . . , M ds )d [M i , M j ],

where [M i , M j ]t B ([M i +M j ]t − [M i −M j ]t )/4 denotes the covariation of M i and M j .

Theorem 2.3.2. (Dubins-Schwarz) Let M be a continuous local martingale for which M0 = 0 al-most surely and limt→∞[M ]t =∞ almost surely. Let σ(t ) = infs : [M ]s > t . Then for all t ≥ 0, σ(t )is an (Fs)s≥0-stopping time. Also, (Fσ(t ))t≥0 is a filtration and Mσ(t ) is a Brownian motion adaptedto (Fσ(t ))t≥0.


Theorem 2.3.3. (Conformal invariance of Brownian motion) Let D be a domain and let (Bt )t≥0 bea Brownian motion started at z ∈ D . If f : D → C is analytic, then there exists a Brownian motionBt in f (D) started at f (z) for which f (Bt ) = B∫ t

0 | f ′(Bs )|2 d s .

Proof. Write z = x+i y , f (z) = u(x, y)+i v(x, y), and Bt = X t +i Yt where X t and Yt are independentscalar Brownian motions. Recall that the real and imaginary parts u and v of an analytic functionf = u + i v satisfy the Cauchy-Riemann equations

∂u

∂x= ∂v

∂y,

∂u

∂y=−∂v

∂x.

By taking further partial derivatives, we see from these equations that u and v are harmonic.Therefore, Itô’s formula gives

du(X t ,Yt ) = ∂u

∂x(X t ,Yt )d X t +

∂u

∂y(X t ,Yt )dYt

+ 1

2

(∂2u

∂x2+ ∂2 y

∂y2

)d t + 1

2

∂2u

∂x∂y(X t ,Yt )d [X ,Y ]t

= ∂u


∂u

∂y(X t ,Yt )dYt .

Similarly, d v(X t ,Yt ) = ∂v/∂x(X t ,Yt )d X t +∂v/∂y(X t ,Yt )dYt . From these expressions we may com-pute the quadratic variation [u(B)]t = [v(B)]t =

∫ t0 | f ′(Bs)|2 d s and the covariation [u(B), v(B)]t = 0.

Let σ(t ) = infs ≥ 0 :∫ s

0 | f ′(Bu)|2 du > t , and define Bt = f (Bσ(t )) = u(Bσ(t ))+ i v(Bσ(t )). By theDubins-Schwarz theorem, Bt is a Brownian motion with respect to the filtration Fσ(t ).

2.4 Applications to complex analysis

In this section we use properties of planar Brownian motion to give simple proofs for several fun-damental results in complex analysis. We begin with a lemma stating a basic property of harmonicfunctions in preparation for a proof of the maximum modulus principle. Proofs of Theorem 2.4.2and 2.4.3 come from [11], while the proof of Theorem 2.4.4 appears in [8].

Lemma 2.4.1. If h is harmonic on D = w ∈C : |w − z| < R, then for all r < R, we have

h(z0) =∫ 2π

0h(z + r exp(iθ))dθ.

Proof. Let τ be the exit time of a planar Brownian motion started at z0 from the disk of radius rcentred at z0. By Theorem 2.2.4, (h(Bt ))t≥0 is a martingale. Apply the optional stopping theoremto find that h(z0) =Ph(Bτ) = ∫ 2π

0 h(z + r exp(iθ))dθ, as desired.

Theorem 2.4.2. (Maximum Modulus Principle) If U ⊂ C is a domain and f : U → C is a noncon-stant analytic function, then | f | has no local maxima in U .

Proof. Suppose for the sake of contradiction that there exists z0 ∈U and ε > 0 for which | f (z0)| ≥| f (z0 + r exp(iθ)| for all 0 < r < ε. By adding a suitable constant to f , we may assume that theimage of z : |z − z0| ≤ ε under f is contained in the right half-plane. Therefore, log f is analyticon z : |z − z0| < ε, from which it follows that the real part log | f | is harmonic. By the previoustheorem, for any 0 < r < ε we have that log | f (z0)| is an average of the values log | f (z0 + r exp(iθ)|.Therefore, log | f | is constant on ∆B z : |z − z0| < ε. This implies that | f | is constant on ∆, whichin turn implies that f is constant on ∆ by the Cauchy-Riemann equations. Since U is connected,f is constant on U .


Theorem 2.4.3. (Fundamental Theorem of Algebra) If p is a nonconstant polynomial, then thereexists z ∈C for which p(z) = 0.

Proof. Suppose that p(z) , 0 for all z ∈ C. Then f = 1/p is an analytic function on C, and sincep →∞ as z →∞, f is bounded. Let Bt be a Brownian motion started at the origin. By Theorem2.2.4, Re f (Bt ) is a martingale. Since Re f is bounded, the martingale convergence theorem impliesthat Re f (Bt ) converges almost surely as t →∞. On the other hand, Re f (C) contains more thanone element, since f is nonconstant. Choose α < β so that infRe f (C) < α < β < supRe f (C). Thesets U1B z : Re f (z) < α and U2B z : Re f (z) > β, are nonempty, disjoint open sets in C. ByTheorem 2.2.5, the Brownian motion visits each of the sets U1 and U2 at arbitrarily large times,so liminfRe f (Bt ) < α < β < limsupRe f (Bt ), almost surely. This contradicts the convergence ofRe f (Bt ).

Theorem 2.4.4. (Schwarz Lemma) Let D = z : |z| < 1 denote the unit disk. If f : D → D is ananalytic function with f (0) = 0, then | f (z)| ≤ |z| for all z. Moreover, if there exists z , 0 for which| f (z)| = |z|, then there exists θ ∈R for which f (z) = e iθz for all z ∈ D .

Proof. Let z0 ∈ D and choose 0 < r < 1 so that z0 is contained in the open disk centred at the originwith radius r . Let S denote the hitting time of the circles of radius r . The function g (z) = f (z)/z iscontinuous at the origin since limz→0 g (z) = f ′(z). Thus g is analytic in D \0 and continuous at 0,and therefore analytic in D . Let Bt be a Brownian motion started at z0, and applying the optionalstopping theorem to g (Bt ) to find

g (z0) = Eg (BS)

Since | f (z)| ≤ 1 for all z ∈ D , we have |g (BS)| = | f (BS)|/|BS | ≤ 1/r . Letting r → 1 gives |g (z0)| ≤ 1.Moreover, if |g (z0)| = 1, then z0 is a local maximum for g . By the previous theorem, this impliesthat g is constant, and |g | = 1. Therefore, there exists θ ∈ R for which g (z) = exp(iθ), so f (z) =z exp(iθ).

For the proof of the following theorem, we will make use of the conformal invariance of Brownianmotion. It comes from [2], where it is stated as an exercise.

Theorem 2.4.5. (Liouville’s theorem) If f : C→ C is a bounded analytic function, then f is con-stant.

Proof. Let Bt be a Brownian motion in C started at the origin. By the previous theorem, f (Bt ) is atime change of a Brownian motion. Since Brownian motion visits every neighbourhood of ∞, theboundedness of f requires that that the time change does not go to ∞. That is,

∫ ∞0 | f ′(Bs)|2 d s <∞

almost surely. We claim that this holds only if f is constant. For if f is nonconstant, then wemay choose a disk D in C whose closure contains no zeros of f . By the open mapping theorem,there exists δ > 0 so that f (z) ≥ δ for all z ∈ D . Define Sn and Tn to be the nth entrance andexit times, respectively, of D . Then Sn and Tn are finite for all n almost surely. By the Markovproperty of Brownian motion, the variables Tn −Sn are i.i.d. Also, each has finite expectation byProposition 2.1.12. By the strong law of large numbers,

∫ ∞0 | f ′(Bs)|2 d x ≥ ∑∞

n=1δ2(Tn − Sn) = ∞

almost surely.

2.5 Applications to planar Brownian motion

Conformal invariance and properties of analytic maps can also be used to answer questions aboutplanar Brownian motion. The latter half of the paper is devoted to carrying out this task for the


question of the Hausdorff dimension of the Brownian frontier. We now present two simpler exam-ples.

Definition 2.5.1. The number of windings around 0 of a path γ : [0, t ] → R2 is defined by (θ(t )−θ(0))/2π, where θ is a continuous function for which the argument of γ(s) is θ(s) for all 0 ≤ s ≤ t .

Theorem 2.5.2. Let Bt be a planar Brownian motion started at (ε,0). Denote by nε the number ofwindings of Bt around the origin before the first time Bt hits the unit circle centred at the origin.For all 0 < ε< 1, the law of 2πnε/logε is the Cauchy distribution.

Proof. Let X t be a Brownian motion started at (logε,0) and stopped at τ = inft ≥ 0 : Re X t ≥ 0.Then Bt is equal in distribution to exp

(Xσ(t )

), where σ(t ) is the time-change given in Theorem

2.3.3. Moreover, the imaginary part of X t is equal in distribution to 2πnε, since Im X t is a continu-ous realisation of the multi-valued function argexp X t which starts at 0 (see Figure 2).

(logε,0)

0

X t

(ε,0)0

Bt

z 7→ exp(z)

Figure 2: The Brownian motion X t started at (logε,0) and its image under theexponential map.

Define x = − logε and St = sup0≤s≤t (Re Xs + x), and observe that τ < t if and only if St < x. Com-bining Propositions 2.1.5(ii) and 2.1.11, we find that the random time τ has the same distributionas x2/Z 2, where Z a standard normal random variable. Also, because τ and Im X are independent,

we may apply the scaling property of Brownian motion to obtain Im Xτd= τ1/2 Im X1 (to see that

this equality holds for all stopping times independent of X , note that it is straightforward for a

stopping times of the form n−1dτ/ne, and take n →∞). Thus Im Xτd= ( x

Z

)(Im X1) = x

(Im X1

Z

). Re-

calling that the quotient of two independent standard normal random variables has the Cauchy

distribution, we have Im Xτd=C x, where C is a Cauchy random variable. Therefore, 2πnε

d=C x ⇒2πnε/logε

d=C .

Given a subset S of the boundary of a domain D , one expects that (under suitable regularity con-ditions on D), a Brownian motion started at a point in D has positive probability of exiting D at apoint in S. We will state and prove this result in the case that D is a Jordan domain. Recall that aJordan domain is defined to be a simply connected domain in C whose boundary is the trajectoryof a simple piecewise-smooth closed curve.

Proposition 2.5.3. Let B be a Brownian motion started at a point z in a Jordan domain D , and letτ denote the hitting time of C\ D . If S ⊂ ∂D has positive arc length, then P(Bτ ∈ S) > 0.

Proof. Since D is a Jordan domain, there exists a conformal map ϕ from D to the unit disk whichsends z to 0 and which extends to a homeomorphism from the closure of D to the closed unit disk[9]. Let B ′ be a Brownian motion in the unit disk with τ′ the hitting time of the unit circle. By con-formal invariance, P(Bτ ∈ S) = P(B ′

τ′ ∈ ϕ(S)). Since ϕ∣∣∂D : ∂D → z : |z| = 1 is a homeomorphism,

ϕ(S) has positive arc length. By rotational symmetry, P(B ′τ′ ∈ϕ(S)) = (the length of S)/(2π) > 0.

3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 11

3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER

3.1 Overview

If A is a bounded subset ofRn , then the Hausdorff dimension is (roughly) the exponent d for whichthe number of balls of radius r needed to cover A scales as 1/r d as r → 0. For a path in R2 whosetrajectory exhibits similar behaviour when viewed on arbitrarily small scales, we may think of theHausdorff dimension of as a measure of how “wiggly” the path is. A smooth path has dimension 1,a space-filling path has dimension 2, and paths like those represented by Figure 5 have dimensionstrictly between 1 and 2, as we shall see. If B is a Brownian motion in Rn for n ≥ 2, then B [0,1] hasHausdorff dimension 2 almost surely (for a proof, see [5]).

The hull of a path γ : [0, t ] → C is defined to be the complement of the unbounded component ofC\γ[0, t ]. Informally, it is “the trajectory γ[0, t ] with the holes filled in.” The boundary of the hull iscalled the frontier. The frontier of a Brownian path is a proper subset of the Brownian path itself,since the path makes excursions to the interior of the hull. In 1982, Mandelbrot conjectured thatthe Hausdorff dimension of the Brownian frontier is 4/3. This conjecture was proved in 2000 byLawler, Schramm, and Werner [6] using the Schramm-Loewner evolution SLEκ, which is a familyof random paths indexed by nonnegative parameter κ. The original proof relies on earlier work [4]that expresses the Hausdorff dimension of the Brownian frontier in terms of the Brownian inter-secting exponent ξ(2,0), which is defined by

P(B [0,TR ]∪B ′[0,T ′

R ] does not disconnect the unit circle from ∞)= R−ξ(2,0)+o(1),

where B and B ′ are independent Brownian motions and TR and T ′R are their exit times from the

disk of radius R. They show that the intersecting exponents are the same for any two randompaths whose laws are completely conformally invariant. They conclude the proof by computingthe intersecting exponents of SLE6 and showing that its law is completely conformally invariant.Rather than following this proof, we will take a more direct route suggested in a recent paper ofBeffara [1]. The proof uses a calculation of the Hausdorff dimension of the SLE paths [1] along withwork of Lawler, Schramm, and Werner which relates the Brownian frontier to SLE8/3 [7]. We beginwith a discussion of Brownian excursions, which prove to be an indispensible tool for making theconnection between the Brownian frontier and SLE8/3.

3.2 Brownian excursions

Let X and W be independent standard Brownian motions in R and R3, respectively, both startedat 0. Define Et = X t + i |Wt |. A Brownian excursion from 0 to ∞ in H is a process which is equal indistribution to (Et )t≥0. A Brownian excursion can be viewed as a Brownian motion conditioned sothat its imaginary part remains positive at all positive times, as we will see in Proposition 3.2.3. Inpreparation for the proof of this theorem, we recall two theorems from stochastic calculus. Proofsmay be found in [2]

Theorem 3.2.1. (Girsanov) Let M be a continuous local martingale on a probability space (Ω,F,P),and suppose that the exponential martingale Zt B exp(Mt − 1

2 [M ]t ) of M is uniformly integrableand that M0 = 0 almost surely. Define a measure P by d P/dP = Z∞. If X is a continuous localmartingale under P, then X − [X , M ] is a continuous local martingale under P.

Theorem 3.2.2. Suppose σ : R→ R and b : R→ R are Lipschitz. If X and X satisfy the stochasticdifferential equation

dX t =σ(X t )dBt +b(σt )dt , (3.2.1)


then X and X have the same distribution.

Proposition 3.2.3. For all z = x + i y ∈ H and R > y , a Brownian motion (Zt )t≥0 = (X t + i Yt )t≥0

started at z and conditioned to hit R+ i R before R is equal in distribution to (Wt + i |Wt |)t≥0, whereW and W are independent standard Brownian motions started at 0 in R and at (y,0,0) in R3, re-spectively.

Proof. For r ≥ 0, let τr denote the hitting time of R+ i r . Define Mt = y−1Yτ0∧τR∧t , and observethat Mt is a bounded nonnegative martingale with M∞ = limt→∞ Mt = R

y 1τ0>τr . We define a new

measure P by d P/dP= M∞, where P is the law of (Bt )t≥0. Under P, we have τ0 > τR almost surely.To apply Girsanov’s theorem, note that Mt is the exponential martingale of

∫ t0 M−1

t dMt . Therefore,the processes (X t )t≥0 and

Bt = Yt −∫ t

0M−1

s d [Y , M ]s

are continuous local martingales under P. Since [X , X ]t = [B ,B ]t = t and [X ,B ] = 0, X and B areindependent P-Brownian motions by Lévy’s characterisation. Moreover, under P,

Yt = y +Bt +∫ t

0

d s

Ys.

Using Itô’s formula with f (x) = |x| in the domain R3 \ 0, we find

|Wt | = |Wo |+3∑

i=1

∫ t

0

W is

|Ws |dW i

s +1

2

∫ t

0

2ds

|Wt |The first term on the right-hand side is y and the second term is a standard Brownian motion, byLévy’s characterisation. Therefore, the process (|Wt |)t≥0 satisfies the same stochastic differentialequation (3.2.1) as Yt under P, with σ = 1 and b(x) = 1/x. Although b is not Lipschitz, we mayreplace b with a Lipschitz function bn which is equal to b up to time τ1/n . By Theorem 3.2.2, thisimplies that for all n, (Yt )t≥0 and (|Wt |)t≥0 have the same law up to time τ1/n . Taking n →∞, wefind that (Yt )t≥0 and (|Wt |)t≥0 have the same law up to time τ0. Moreover, the law of Yt under Pconditioned on the event τR < τ0 is the same as the law of Yt under P, because Proposition 2.1.12gives

P(Zt ∈ A) = E(M∞1Zt∈A) =R

yP(Zt ∈ A and τ0 > τR ) = P(Zt ∈ A and τ0 > τR )

P(τ0 > τR ),

for all A ∈ B(C ([0,∞),C). The last quotient is the elementary conditional probability of A giventhat Zt hits R + iR before R.

Definition 3.2.4. A bounded set K ⊂H for which K ∩H= K andH\K is simply connected is calleda compact H-hull. The set of compact H-hulls is denoted Q. The set Q+ is defined to contain thecompact H-hulls whose intersection with the negative ray (−∞,0] is empty. The set Q− is definedanalogously with the positive ray [0,∞). We define Q± =Q+∪Q−.

The compact H hulls are in one-to-one correspondence with a certain collection of conformalmaps, as shown by the next proposition. For a proof, see [5].

Proposition 3.2.5. For each compact H-hull K , there exists a unique surjective conformal mapgK :H\ K →Hwhich satisfies gK (z)− z → 0 as |z|→∞.

The next theorem provides a further connection between compactH-hulls K and their associated


maps gK . We use the notation X [a,b) for the trajectory X t : a ≤ t < b of a process (X t )t≥0 on thetime interval [a,b).

Theorem 3.2.6. Let (Et )t≥0 be a Brownian excursion and let K ∈Q±. We have

P(E [0,∞)∩K =;) = g ′K (0).

Proof. First observe that by the Schwarz reflection principle, gK can be extended to an analyticfunction in (H \ K ) ∪ N for some neighbourhood N of the origin (because the boundary of H \K is “locally analytic” near the origin, see [5]). Therefore, g ′

K (0) exists. By the continuity of g

on N ∩H, we have g (N ∩R) ⊂ R. This implies that g ′K (0) is real, which in turn implies g ′

K (0) =limw→0 Im gK (w)/Im w .

Now let τR be the hitting time of R+ i R. Let t > 0 and let B be a Brownian motion started at Et . Bythe previous proposition,

P(E [t ,τR )∩K =;) = P(B hits R+ i R before R∪K )

P(B hits R+ i R before R). (3.2.2)

A Brownian motion B started at z ∈Hhas probability Im(z)/R of hittingR+i R beforeR, by Proposi-tion 2.1.12. The probability that B hitsR+i R beforeR∪K is equal to the probability that a Brownianmotion started at gK (z) hits gK (R+ i R) before R, by conformal invariance. Since gK (z)− z → 0 asz →∞, we may choose R large enough that gK (R+ i R) lies in the strip z : R −1 < Im z < R +1.Therefore for R > 1,

Im gK (z)

R +1<P(B hits R+ i R before R∪K ) < Im gK (z)

R −1Substitute into (3.2.2) and take R →∞ to find

P(E [t ,∞)∩K =;) = E(

Im gK (Et )

ImEt

).

Taking t → 0 and applying dominated convergence on the right-hand side gives P(E [0,∞)∩K =;) = g ′

K (0).

3.3 Reflected Brownian excursions

Fix θ ∈ (0,π) and consider a C-valued process B = X + i Y started at a point z in the wedge W (θ)Br e iϕ : r > 0 and 0 <ϕ< π−θ. We will define a new process B = X + i Y taking values in W (θ) byreflecting B horizontally off the ray r e i (π−θ) : r > 0 and stopping it when it hits the real line. Moreprecisely, let c =−cotθ, let Yt = Yt , and let

X t = X t + sups≤t

((cYt −X t )∨0

),

which is the unique process for which X t ≥ cYt and `t = X t −X t satisfies

(i) `t is nondecreasing and continuous, and(ii) The Stieltjes measure d`t is supported on the set t ≥ 0 : X t = cYt .

See [10] for a proof of uniqueness. The relationship between X t , cYt , and X t is illustrated in Figure3 below with toy functions X t and cYt .

Proposition 3.3.1. Let Φ : W (θ) →W (θ) be a conformal map which fixes the origin and for which|Φ(z)− z| : z ∈ W (θ) is bounded, and let (Bt )0≤t≤τR+ be a reflected Brownian motion started atz ∈W (θ). Then (Φ(Bt ))0≤t≤τR+ is a time change of a reflected Brownian motion started atΦ(z).

Proof. Define the ray ρ = r e i (π−θ) : r > 0. The hypotheses ensure thatΦ′(z) > 0 for all z ∈ ρ. (Note


X t

cYt

X t

Figure 3: The reflected process (X t )t≥0.

that the derivative exists because the boundary of W (θ) is straight in a neighbourhood of z ∈ ρ).Also, since d`t is supported on Bt ∈ ρ, the process ˜

t =∫ t

0 Φ′(Bs)d`s is continuous, nondecreas-

ing, and has its Stieltjes measure d ˜t supported on Φ(B) ∈ ρ. Therefore, it suffices to show that

Φ(B)− ˜ is a time change of Brownian motion up to its exit time from H. Write z = x + i y andΦ(x + i y) = u(x, y)+ i v(x, y). By Itô’s formula, we have

du(X ,Y ) = ∂u


∂u

∂y(X t ,Yt )dYt −d ˜

t

= ∂u

∂x(X t ,Yt ) (d X t +d`t )+ ∂u

∂y(X t ,Yt )dYt −d ˜

t

= ∂u


∂u

∂y(X t ,Yt )dYt ,

so the real part ofΦ(B)− ˜ is a local martingale. It follows from Lévy’s characterisation thatΦ(B)− ˜

is a time change of Brownian motion, as in the proof of Theorem 2.3.3.

θ

W (θ)

0 0

Hbθ(z) = zπ/(θ−π)

Figure 4: A sample Brownian excursion reflected off the ray r e3πi /8 : r ≥ 0 inW (θ) and its image under bθ.

Observe that that bθ(z) = zπ/(π−θ) is a conformal map of W (θ) ontoHwhich fixes the origin.

Definition 3.3.2. Let E be a Brownian excursion and let E be the reflection of E in the wedge W (θ).A reflected Brownian excursion with angle θ is a process E which is equal in law to bθ(E).

Theorem 3.3.3. Let E be a reflected Brownian excursion with angle θ. Then for all K ∈Q+, we have

P(E [0,∞)∩K =;) =Φ′K (0)1−θ/π,

where ΦK (z) = gK (z)− gK (0) is the unique conformal isomorphism of H \ K → H which satisfiesΦK (0) = 0 andΦK (z)/z → 1 as z →∞.

Proof. For r ≥ 0, let τr the hitting time of R+ i r . By Proposition 3.2.3, for all r > 0, the Brownian


excursion started at Eτr and stopped at τR has the same law as a reflected Brownian motion startedat Eτr , conditioned on the set τr < τ0 and stopped at τR . Therefore, by Proposition 3.3.1, we maycompute as in the proof of Proposition 3.2.6,

P(E [τR ,∞)∩K =;) = E[

ImΦ(Eτr )

Im Eτr

], (3.3.1)

for all r > 0, where Φ is the unique conformal map from W (θ)\b−1(K ) →W (θ) that fixes the originand satisfies Φ(z)/z → 1 as z →∞. Since b−1

θΦK bθ satisfies these requirements, Φ= b−1

θΦK bθ.

As z → 0, we have

Φ(z) = b−1θ (ΦK (bθ(z)))

=(ΦK (zπ/(θ−π))

)1−θ/π

=(Φ′

K (0)zπ/(θ−π) +O(|z2π/(θ−π)|))1−θ/π

,

so Φ′(0) =Φ′K (0)1−θ/π. Therefore, taking the limit as r → 0 in (3.3.1) and applying dominated con-

vergence, we find

P(E [0,∞)∩K =;) =Φ′K (0)1−π/θ.

Let us make note of the case θ = 3π/8.

Corollary 3.3.4. If K ∈Q+, and E is a reflected Brownian excursion with angle θ = 3π/8, then

P(E [0,∞)∩K =;) =Φ′K (0)5/8. (3.3.2)

3.4 Loewner’s theorem

It turns out that there exists a simple random curve γ from 0 to ∞ inHwhich satisfies the same law(3.3.2) as a reflected Brownian excursion with angle 3π/8. Moreover, the Hausdorff dimension ofthis curve is known. We will use this observation to compute the almost-sure Hausdorff dimensionof the Brownian frontier. The curve γ is an example of a class of curves called Schramm-Loewnerevolutions (SLE). We will briefly sketch the framework in which SLE is defined. Proofs may befound in [5] or [8]. Let K be a compact H-hull and let gK be the unique conformal map of H \ Konto H for which gK (z)− z → 0 as |z| →∞. Then there exists a unique aK ∈ R for which gK (z) =z + aK /z +O(|z|−2) as |z| → ∞. The quantity aK is denoted hcap(K ) and is called the half-planecapacity of K . Define rad(K ) to be the radius of the smallest circle containing K whose centre is inR. Let (Kt )t≥0 be an increasing family of compact H-hulls, and define Kt ,t+h = gKt (Kt+h \ Kh). Thefamily (Kt )t≥0 is said to satisfy the local growth property if Kt ,t+h → 0 as h → 0, uniformly as t rangesover compact subsets of R+. If (Kt )t≥0 has the local growth property, then for all t ,

⋂h>0 Kt ,t+h

contains a single point, which we will define to be ξt . The function ξ :R+ →R is called the Loewnertransform of (Kt )t≥0, and it is continuous. Loewner’s theorem describes a correspondence betweenξ and the differential equation

∂

∂tg t (z) = 2

g t (z)−ξt, g0(z) = z,

which is called Loewner’s equation. Observe that the solution of Loewner’s equation does not nec-essarily exist for all t ≥ 0, since the denominator can go to zero. For each z ∈C, we define Tz to bethe supremum of the times up to which a solution exists. For a proof of Loewner’s theorem, see[8].

Theorem 3.4.1. (Loewner)


0κ= 8/3

0κ= 4

0κ= 6

Figure 5: Sample paths of numerical approximations of SLEκ for κ = 8/3, 4, and6.

(i) Suppose that (Kt )t≥0 is an increasing family of compact H-hulls satisfying the local growthproperty. Suppose further that (Kt )t≥0 has been parametrised so that hcap(Kt ) = 2t . Thenfor each z ∈H, the solution of the Loewner equation up to time Tz is given by t 7→ gKt (z).

(ii) Let (ξt )t≥0 be a continuous, real-valued process. For z ∈H, define g t (z) to be the solution ofLoewner’s equation up to time Tz , and define Kt = z : Tz ≤ t . Then (Kt )t≥0 is an increasingfamily of compact H-hulls satisfying the local growth property, hcap(Kt ) = 2t , the Loewnertransform of (Kt )t≥0 is ξ, and gKt = g t for all t ≥ 0.

In the context of part (ii) of Loewner’s theorem, ξ is called the driving function. The Schramm-Loewner evolution is obtained by taking the driving function of Loewner’s equation to be a Brow-nian motion with a diffusivity κ ≥ 0, i.e. ξt =

pκBt , where Bt is a standard Brownian motion. It

can be shown [5] that the complement Kt of Ht is the hull of a unique path γ. The random pathγ is called SLEκ. Its behaviour exhibits phase transitions as κ increases, as stated in the followingproposition (for a proof, see [5]).

Proposition 3.4.2. Let γ be an SLEκ. Then

1. If 0 ≤ κ≤ 4, γ is a simple path, almost surely.2. If 4 < κ< 8, γ is neither simple nor space-filling, almost surely.3. If 8 ≤ κ, γ is almost surely space-filling.

As we shall now show, SLE curves inherit the scaling property from Brownian motion.

Proposition 3.4.3. Let γ be an SLEκ and let c > 0. Then (cγt/c2 )t≥0 is also an SLEκ.

Proof. For each t ≥ 0, let g t be the conformal map corresponding to γt . Define g t (z) = cg t/c2 (z/c),which is the family of conformal maps corresponding to cγt/c2 . Then

∂g t (z)

∂t= 1

cg t/c2 (z/c) = 2/c

g t/c2 (z/c)− pκBt/c2

= 2

g t (z)− pκcBt/c2

,

and the second term in the denominator is a Brownian motion by Proposition 2.1.5(ii).


3.5 Restriction property of SLE8/3

Let us consider the image of a growing family of compact H-hulls under a conformal map. Inparticular, let (Kt )t≥0 be a family of compact H-hulls satisfying the local growth property, let ξt =⋂

h>0 Kt ,t+h be the driving function associated with Kt , and let (g t )t≥0 be the corresponding familyof conformal maps g t :H\Kt →H. LetΦ(z) = a0+a1z+a2z2+·· · be a conformal map defined in anH-neighbourhood U of ξ0. Note that for Φ(U ) ⊂H, we must have a0, a1, a2, . . . ∈ R and a1 > 0. LetT = supt ≥ 0 : Kt ⊂U . For 0 ≤ t < T , define K ∗

t =Φ(Kt ), a∗t = hcap(K ∗

t ), and let g∗t be the unique

conformal mapH\ K ∗t →Hwith g∗

t (z)− z → 0 as z →∞. Finally, letΦt = g−1t Φ g∗

t .

Proposition 3.5.1. With the preceding notation, we have

(i) (K ∗t )0≤t<T is a family of compactH-hulls satisfying the local growth property,

(ii) (ξ∗t )0≤t<T is the Loewner transform of (K ∗t )0≤t<T ,

(iii) t 7→ a∗t has derivative a∗

t = 2Φ′t (ξt )2, and

(iv) (t , z) 7→Φt (z) is differentiable in both coordinates with

Φt (ξt ) =−3Φ′′t (ξt ), and (3.5.1)

Φ′t (ξt ) = Φ

′′t (ξt )2

2Φ′t (ξt )

− 4

3Φ′′′

t (ξt ). (3.5.2)

Sketch of proof. (i) and (ii) are easy to check. For (iii), observe that a∗t+h − a∗

t = hcap(g∗t (K ∗

t+h \K ∗

t )) ≈ hcap(ξ∗t +Φ′t (ξt ))(g t (Kt+h −Kt )− ξt ) = Φ′

t (ξt )2 hcap(g t (Kt+h −Kt ) = 2hΦ′t (ξt )2. To derive

(iv), assume first that t = 0. Write ft for g−1t . Differentiate ft (g t (z)) = z with respect to t to find that

ft (z) = −2 f ′t (z)

z −Ut.

Also, note that g∗0 (z) = a∗

tz−ξ∗0

from Loewner’s equation, and apply the chain rule:

Φ0(z) = g∗0 (Φ0( f0(z)))+ (g∗

0 )′(Φ0( f0(z)))Φ′0( f0(z)) f0(z)

= 2Φ′0(ξ0)2

Φ0(z)−Φ0(ξ0)− (g∗

0 )′(Φ0( f0(z)))Φ′0( f0(z)) f ′

0(z)2

z −U0

= 2Φ′0(ξ0)2

Φ0(z)−Φ0(ξ0)− 2Φ′

0(z)

z −ξ0.

For 0 < t < T , we can apply the map g t to (Ks)t≤s<T and g∗t to (K ∗

s )t≤s<T to reduce to the case t = 0.Therefore

Φt (z) = 2Φ′t (ξt )2

Φt (z)−Φt (ξt )− 2Φ′

t (z)

z −ξt. (3.5.3)

Letting z → ξt gives (3.5.1). Since the RHS of 3.5.3 is continuous, we can find Φ′(z) by differentiat-ing with respect to z to find

Φ′t (z) = −2Φ′

t (ξt )2Φ′t (z)

(Φt (z)−Φt (ξt ))2+ 2Φt (z)

(z −Ut )2− 2Φ′′

t (z)

z −ξt, (3.5.4)

which gives (3.5.2) by letting z → ξt .

Proposition 3.5.2. The process Mt = 1t<τAΦ′t (ξt )α satisfies

d Mt

αMt= Φ

′′t (ξt )

Φ′t (ξt )

pκdBt +

[((α−1)κ+1

2

)Φ′′

t (ξt )2

Φ′t (ξt )2

+(κ

2− 4

3

)Φ′′′

t (ξt )

Φ′t (ξt )

]d t . (3.5.5)


Proof. We apply Itô’s formula to find

d Mt

αMt= Φ

′t (ξt )

Φ′t (ξt )

dt + Φ′′t (ξt )

Φ′t (ξt )

pκdBt +

1

2

[(α−1)

Φ′′t (ξt )2

Φ′t (ξt )2

+ Φ′′′t (ξt )

Φ′t (ξt )

]κdt ,

Equation (3.5.5) follows by substituting the expression in (3.5.2) for Φ′t (ξt ).

For the following theorem we present a proof sketch given in [8]. See [5] for more details.

Theorem 3.5.3. Let K ∈Q± and let γ be an SLE8/3. Then

P(γ[0,∞)∩K =;) =Φ′K (0)5/8,

where ΦK (z) = gK (z)− gK (0) is the unique conformal isomorphism of H \ K → H which satisfiesΦK (0) = 0 andΦK (z)/z → 1 as z →∞.

Proof sketch. Let D = H \ K , and for ε > 0, let Dε denote the set of all points in H whose distancefrom K is at least ε. Define g t , g∗

t , and Φt as in Proposition 3.5.1 with Φ=ΦDε . Let τ be the hittingtime of R∪K and Mt = 1t<τΦ

′t (ξt )α. Observe that choice κ = 8/3 and α = 5/8 makes (Mt )t≥0 a

local martingale, by Proposition 3.5.2. Since 0 ≤ Φ′t (ξt ) ≤ 1, (Mt )t≥0 is a martingale. Note that by

Proposition 3.2.6,

Φ′t (ξt ) =P(E [0,∞) ⊂ g t (Dε)), (3.5.6)

where E is a Brownian excursion started at ξt . Our goal is to show that Mt approaches the indicatorof τ=∞ = γ[0,∞)∩K =; as t →∞. Note that as t →∞, g t (H\Dε) gets smaller and farther fromthe origin, which (when made rigorous) implies Mt → 1 on the set τ =∞. Also, on τ <∞, γτlies on a ball B at least half of which is contained in H \ Dε. Therefore, there exists p > 0 for whicha Brownian motion started at any point in B ∩H has probability greater than p of exitingH\γ[0, t ]on the boundary (−∞,γτ) and also probability greater than p of exiting on (γτ,∞). It follows thatgτ(B ∩H) is contained in a proper cone with vertex ξτ. By (3.5.6), this impliesΦ′

τ(ξτ) = 0. Thus

Φ′Dε

(0)5/8 = M0 = EMτ = 1 ·P(γ⊂ Dε)+0 ·P(γ∩H\ Dε ,;) =P(γ⊂ Dε).

Taking ε→ 0 gives the result.

Remark 3.5.4. The use of the term restriction for the property described in Theorem 3.5.3 comesfrom the fact that the theorem gives a quick proof of the restriction property of SLE8/3. See [8] formore details.

Remark 3.5.5. By Corollary 3.3.4 and Theorem 3.5.3, SLE8/3 has the same law as the right bound-ary of an RBE with angle 3π/8. We will use this observation to show that that they have the sameHausdorff dimension, from which it will follow that the dimension of the Brownian frontier is equalto the dimension of SLE8/3.

3.6 Hausdorff dimension

We now give a rigorous definition of the Hausdorff dimension and prove some of its basic proper-ties. For α≥ 0, ε> 0, and A ⊂C, define

Hαε (A) = inf

∞∑n=1

(diamDn)α : A ⊂⋃n

Dn ; ∀n,diamDn ≤ ε

.

As ε → 0, the set of coverings Dn∞n=1 decreases, so Hαε (A) increases. Therefore, limε0 Hα

ε (A)exists in [0,∞].

Proposition 3.6.1. Hα(A) <∞⇒ Hβ(A) = 0 for all β>α.


Proof. Calculate

Hβε (A) = inf

∞∑n=1

(diamDn)β : A ⊂⋃n


= inf

∞∑n=1

(diamDn)β−α(diamDn)α : A ⊂⋃n


≤ εβ−α inf

∞∑n=1

(diamDn)α : A ⊂⋃n


= εβ−αHαε (A) → 0 as ε→ 0.

Proposition 3.6.1 and establishes the existence of α0 such that Hα(A) is equal to ∞ on the interval(0,α0) and 0 on the interval (α0,∞). This critical valueα0 is defined to be the Hausdorff dimensionof A, and is denoted dimH (A).

Lemma 3.6.2. Hα(A) = 0 if and only if for all sequences εm → 0+ and δm → 0+, there exist sets Um, j

for which diamUm, j < εm for all j and∑

j (diamUm, j )α < δm .

Proof. Since Hαε (A) increases as ε decreases, Hα(A) = 0 if and only if Hα

ε (A) = 0 for all ε > 0. Theequivalence of

Hαε (A) = 0 for all ε> 0

and the existence of Um, j follows from the definition of a limit.

Proposition 3.6.3. If f : A → C is b-Hölder continuous (that is, | f (z)− f (w)| ≤ c|z − w |b for allz, w ∈C), then dimH ( f (A)) ≤ b−1 dimH (A).

Proof. (From [5]) Let α> dimH (A), let εm → 0+ and δm → 0+, and let Um, j satisfy the conditions inLemma 3.6.2. Then the sets Um, j = f (V ∩Um, j ) satisfy diamUm, j ≤ cεb

m and∑

j (diamUm, j )α/b ≤cα/b ∑

j (diamUm, j )α < cα/bδm . Thus dimH f (A) ≤α/b for allα> dimH A, which implies dimH ( f (A)) ≤b−1 dimH (A).

Proposition 3.6.4. If A =∞⋃

n=1An , then dimH A = supn dimH An .

Proof. It is clear that dimH A ≥ supn dimH An . Conversely, let α > supn dimH An . Let Un,m, j sat-isfy Un,m, j < 2−m−n for all j and

∑j diam(Un,m, j )α < 2−m−n for all m,n. Let Um, j =

⋃n Un,m, j . By

the triangle inequality, diamUm, j < 2−m for all j . Also,∑

j (diamUm, j )α < ∑n 2−m−n = 2−m . Thus

dimH A ≤α, as desired.

Definition 3.6.5. We say that a process (X t )t≥0 satisfies the scaling property if (c X t/c2 )t≥0 has thesame distribution as (X t )t≥0, for all c > 0.

Proposition 3.6.6. Suppose that (X t )t≥0 satisfies the scaling property. For 0 < s < t <∞, the ran-dom variable dimH X [s, t ] has the same distribution as dimH X [0,∞).

Proof. Write X (0,∞) =⋃q X [qs, qt ], where the union ranges over all rationals q > 0. By the scaling

hypothesis, the distribution of the dimension of X [qs, qt ] does not depend on q . By Proposition3.6.4, the distribution of dimH X [qs, qt ] is equal to that of X (0,∞).


Proposition 3.6.7. Suppose that (X t )t≥0 satisfies the scaling property, and let d ≥ 0. Then

dimH X [0,∞) = d almost surely

implies that dimH X [σ,τ] = d almost surely for all random times σ and τ with 0 <σ< τ<∞.

Proof. Suppose first σ and τ take on only countably many values, say sn∞n=1 and tn∞n=1. Then

P(

dimH (γ[σ,τ] = d))= ∞∑

m,n=1P

(dimH X [sm , tn] = d and (σ,τ) = (sm , sn))

)=

∞∑m,n=1

P((σ,τ) = (sm , tn)

)= 1.

For general σ and τ, define σn B n−1dnσe and τn B n−1bτnc. Since X [σ,τ] = ⋃n X [σn ,τn], the

result follows from Proposition 3.6.6.

Remark 3.6.8. To make statements about distributions of Hausdorff dimensions of random setsrigorous, we need a σ-algebra on a collection of subsets of C for which the map which sends aset to its Hausdorff dimension is measurable. It is possible to do this, for example, on the set ofcompact subsets of C by taking the Borel sets corresponding to the Hausdorff metric

d(A,B) =(supa∈A

infb∈B

|a −b|)∨

(supb∈B

infa∈A

|a −b|)

.

In Section 3.8, we will define an ad hoc σ-algebra which respects taking the dimension of theboundary rather than the dimension of the set itself. Since we will apply Propositions 3.6.6 and3.6.7 only to simple curves X (namely SLE8/3), this σ-algebra will suffice for our purposes.

3.7 Hausdorff dimension of SLE

A proof of the following theorem may be found in [1]. We will present a heuristic derivation givenin [3]. Observe that for κ ≥ 8, the dimension of SLEκ is trivially 2, since the path is space filling.Also, SLE0 is given by the simple formula γ(t ) = 2i

pt , so the dimension of SLE0 is 1. Between κ= 0

and κ = 8, it turns out that that the relationship between κ and the Hausdorff dimension of SLEκis linear.

Theorem 3.7.1. The Hausdorff dimension of SLEκ is almost surely 2∧ (1+ κ

8

).

Heuristic proof of upper bound. We begin with a general observation about random sets. Recallthat f (ε) = Θ(g (ε)) as ε→ 0 if f is bounded above and below by a constant multiple of g as ε→0. It takes Θ(ε−2) balls of radius ε to cover a given (bounded) region in R2, and of these Θ(ε−d )are needed to cover K . Therefore, we expect that the probability that an ε-neighbourhood of zintersects K should take the form ε2−d f (z).

As previously observed, the theorem is trivially true when κ≥ 8. So fix 0 ≤ κ< 8, and let P (ζ,ζ,ε, a)denote the probability that an SLEκ path γ started at a visits the ε-neighbourhood of ζ ∈H. We willderive an expression for P by showing that it solves a certain partial differential equation. To thisend, imagine evolving the Loewner flow for an infinitesimal time dt . Under the conformal mapgdt , a maps to a′ = a + p

κdBt , ζ maps to ζ′ = ζ+ 2dtζ−a (from the Loewner differential equation),

and ε maps to ε′ = ε|g ′dt (ζ)| = ε− 2εRe((ζ− a)−2)dt . To obtain the last expression, differentiate

ζ 7→ ζ+ 2dtζ−a and observe that for small z, |1− z| ≈ 1−Re z. Recall that that gdt (γ) is an SLEκ started

at a′. Therefore, an SLEκ started from a′ visits visit the ε′-neighbourhood of ζ′ if and only if γ visits


the ε-neighbourhood of ζ. So P satisfies

P (ζ,ζ,ε, a) = E[

P

(ζ+ 2dt

ζ−a,ζ+ 2dt

ζ−a,ε−2εRe

((ζ−a)−2)dt , a + p

κdBt

)]Taylor expand the right-hand side, to first order in the first three arguments and to second orderin the fourth argument. Simplify using EdBt = 0 and E(dBt )2 = d t , and subtract P (ζ,ζ,ε, a) fromboth sides to obtain (

2

ζ−a

∂

∂ζ+ 2

ζ−a

∂

∂ζ+ κ

2

∂2

∂a2−2 Re

((ζ−a)−2)ε ∂

∂ε

)P = 0.

To simplify this differential equation, let a = 0 without loss of generality. Also, switch to Cartesiancoordinates ζ= x + i y and use the identities

∂

∂x= ∂

∂ζ+ ∂

∂ζ, and

∂

∂y= i

∂

∂ζ− i

∂

∂ζ

to rewrite

2

ζ

∂

∂ζ+ 2

ζ

∂

∂ζ= 2x

x2 + y2

∂

∂x− 2y

x2 + y2

∂

∂y.

Finally, observe that P (x, y,ε, a) = P (x −a, y,ε,0), which implies ∂P∂a =−∂P

∂x . Altogether, we find(2x

x2 + y2

∂

∂x− 2y

x2 + y2

∂

∂y+ κ

2

∂2

∂x2− 2(x2 − y2)

(x2 + y2)2ε∂

∂ε

)P = 0 (3.7.1)

We expect that P scales as some power of ε/|ζ|, and by scale invarance we expect that its depen-dence on ζ is a function of the argument θ of ζ alone. It turns out that a power of sinθ works. Write(

εpx2+y2

)2−d (y2

x2+y2

)αas ε2−d yd−2−2β(x2 + y2)β and substitute in (3.7.1) to obtain

−y−2+d−2β(x2 + y2)−2+βε2−d ((κ−8−2κβ)βx2 + (4d −8−κβ)y2)= 0.

Solving the system

κ−8−2κβ= 0,

4d −8−κβ= 0

gives d = 1+ κ8 , as desired.

It is worth emphasizing that the ideas in this derivation are only sufficient to give an upper boundon the dimension, since a priori the form ε2−d f (z) could result from some SLE sample paths in-tersecting many balls, whereas ‘most’ paths intersect fewer [1]. Estimates on the probability of thepath intersecting two arbitrary disks can be used to prove dimH (SLEκ) ≥ 2∧ (1+κ/8).

3.8 Hausdorff dimension of the Brownian frontier

Let us say that a closed set F ⊂ H is left-filled if its intersection with the real line is (−∞,0] andits complement H \ F is simply connected. Let Ω+ denote the set of all such F , equipped with theσ-algebra Σ+ generated by E= SK : K ∈Q+, where SK B F : F ∩K =;. Observe that the law ofF is determined by the values of P(F ∩K =;) for K ∈Q+, since SK1 ∩SK2 = SK1∪K2 shows that E is aπ-system. Denote by ∂r F the right boundary ∂(H\ F ) \ (0,∞) of the filling F .


Proposition 3.8.1. The map F 7→ dimH (∂r F ) from (Ω+,Σ+) to ([0,2],B([0,2]) is measurable.

Proof. Define U to be the set of all balls in C which intersect H, have a rational radius, and whosecentres has rational coordinates. Enumerate the elements U1,U2, . . . of U. Let B ∈ U, and defineE(B) = F : F ∩B , ;. We will show that E(B) ∈ Σ+. First, observe that if V is a bounded simplyconnected open set for which R∩ ∂(H∩V ) is a nonempty subset of (0,∞), then E(V ) ∈ Σ+. Tosee this, approximate V with a countable sequence of compact subsets of V . Define P to be thecollection of all finite unions of elements of U. To see that P is countable, observe that

P=∞⋃

n=1A : A is a union of sets in U1,U2, . . . ,Un.

Enumerate the elements P1,P2, . . . of P. If B intersects ∂r F , then there exists P ∈P for which P \ Bis connected and ∂r F does not intersect the closure of any of the other balls whose union is P [9].Therefore,

E(B) =∞⋃

n=1E(Pn) \ E(Pn \ B),

which shows that E(B) ∈Σ+.

Observe that every D ⊂H is contained in an element of U with diameter at most 2diamD . There-fore, if we define for A ⊂C,

Hα(A) = limk→∞

inf

∞∑n=1

diam(Dn)α : A ⊂∞⋃

n=1Dn , diamDn ≤ 1/k, Dn ∈U for all n

,

we have 2−αHα(A) ≤ Hα(A) ≤ Hα(A). In particular, Hα is nonzero if and only if Hα is nonzero, sodimH (A) = infα : Hα(A) = 0. Writing

Hα(∂r F ) = limk→∞

∞∑n=1

diam(Un)α1E(Un )1diamUn≤1/k

shows that (α,V ) 7→ Hα(V ) is measurable, and finally

dimH (∂r F ) = infα∈Q+

α1Hα(∂r F )=0

shows that F 7→ dimH (∂r F ) is measurable.

Proposition 3.8.2. The right boundary of a reflected Brownian excursion with angle 3π/8 hasHausdorff dimension 4/3 almost surely.

Proof. By Proposition 3.8.1, S B F : dimH (∂r F ) = 4/3 ∈ Σ+. Define µ1 to be the law of the left-filling of SLE8/3, and define µ2 to be the law of the left-filling of the reflected Browian excursionwith angle 3π/8. By Theorem 3.7.1, µ1(S) = 1. By Corollary 3.3.4 and Theorem 3.5.3, we haveµ1 =µ2 on the π-system E which generates Σ+. Therefore, µ2(S) =µ1(S) = 1.

Definition 3.8.3. Define the σ-algebra A on the set of subsets of C to be the one generated by setsof the form

A : A∩F =; : F ⊂ C is closed and C\ F is simply connected

.

Here C denotes the extended complex plane C∪ ∞. Define the frontier fr A of a set A ⊂ C to bethe boundary of the complement of the unbounded component of C\ A.


A

∂l ,x

x

a(A, x)

b(A, x)

Figure 6: A compact set A and its left boundary ∂l ,x .

The σ-algebra A is chosen so that the law of the frontier of a set is determined by the probabilitiesP(A∩F =;), as shown by the next proposition.

Proposition 3.8.4. The map A 7→ dimH (fr A) is A-measurable.

Proof. Follow the proof of Proposition 3.8.1, with the modification that U is defined to be the setof all balls in C with rational radius and centre either at ∞ or at a point with rational coordinates.(Recall that a ball centred at ∞ is a set of the form

z : z > r−1

for some r > 0).

Definition 3.8.5. Let A be a compact set, and let x ∈R. For z ∈C and θ ∈R, define the ray

ρ(z,θ) = z + r e iθ : r ≥ 0,

and define the translation σξ(z) = z − ξ. Define a(A, x) = x + i supIm z : Re z = x and z ∈ A andb(A, x) = x + i infIm z : Re z = x and z ∈ A. Choose y large enough that σ−i y (A) ⊂H, let ξ= x + i y ,and define

∂r,x A = A∩(σ−1ξ ∂r σξ

(A∪ρ

(a(A, x),π/2

)∪ρ

(b(A, x),−π/2

))),

and define ∂l ,x analogously (see Figure 6).

Proposition 3.8.6. The frontier of a compact set A can be written as

fr A = (∂l ,x A

)∪ (∂r,x A

).

Proof. A point z is in the frontier of A if and only if z is “connected to ∞” in C \ A. More pre-cisely, z ∈ fr A if and only if there exists a path γ : [0,1] for which γ(0) = z, γ(1) is in a neighbour-hood of ∞ contained in C \ A, and γ(0,1) is a subset of the unbounded component of C \ A. Bycontinuity, γ(0,1) ⊂ C \ A implies that γ(0,1) is a subset of the unbounded component of C \ A.Since any point on the right or left boundary is connected to ∞, we have

(∂l ,x A

)∪ (∂r,x A

) ⊂ fr A.Conversely, if z ∈ fr A, then let γ be a path connecting z to ∞ in C \ A. If γ does not intersect

ρ(a(A, x),π/2

)∪ ρ

(b(A, x),−π/2

), then z is on either the left or the right frontier. If γ does in-

tersect ρ (a(A, x),π/2)∪ρ (b(A, x),−π/2), then without loss of generality suppose that γ intersectsρ (a(A, x),π/2). Let t0 be the first time γ intersects ρ (a(A, x),π/2), and without loss of generality,assume that γ(t ) approaches ρ (a(A, x),π/2) from the right as t → t0. By compactness, there existsε> 0 for which the strip

z : x < Re z < γ(t0 −ε) and Im z > Im a(A, x)

lies in C\ A. Define a new path γ∗ which is equal to γ up to time t0 −ε/2, follows a vertical ray out


to a neighbourhood of ∞ in C\ A on the interval [t0 −ε/2, t0 +ε/2], and follows a circular arc downto R+ on the interval [t0 +ε/2,1].

Lemma 3.8.7. Suppose that (Kt )t≥0 is a family of random compact subsets of C for which Ktd=p

tK1 for all t > 0. Let τ be a random time whose range is countable. Then dimH Kτ = d almostsurely if and only if dimH K1 = d almost surely.

Proof. Suppose that dimH K1 = d almost surely. By the scaling hypothesis, the distribution ofdimH Kt does not depend on t , so dimH Kt = d almost surely for all t > 0. Enumerate the elementsin the range of τ as tn∞n=1. Then

P(dimH Kτ = d) =P( ∞⋃

n=1

(dimH Ktn = d∩ τ= tn

))=

∞∑n=1

P(dimH Ktn = d∩ τ= tn

)=

∞∑n=1

P(τ= tn) = 1.

Conversely, suppose that dimH Kτ = d almost surely. If Ω is written as a disjoint union⋃

nΩn andA is an event whose probability is less than 1, then there exists a natural number n for whichP(A∩Ωn) <P(Ωn). So if P(dimH K1 , d) > 0, we have

P(dimH Kτ = d) =∞∑

n=1P

(dimH Ktn = d∩ τ= tn

)<

∞∑n=1

P(τ= tn) = 1,

a contradiction.

To make use of the preceding lemma, we define what we will call a neighbourhood rational randomtime, which is a rational-valued random time τwhich stops a continuous process X in an arbitraryneighbourhood of Xσ for any real-valued random time σ. In exchange for the advantage havinga countable range, a neighbourhood rational random time has the disadvantage that it is not astopping time.

Definition 3.8.8. Let (X t )t≥0 be a continuous random process inC, letσ be a random time, and letr be a positive random variable. The neighbourhood rational random time associated with σ andr is defined as follows. Of the rational numbers t >σ with smallest denominator for which X [σ, t ]is contained in the ball of radius r centred at Xσ, let the neighbourhood rational random time bethe smallest.

Theorem 3.8.9. If B is a Brownian motion in C, then dimH frB [0,1] = 4/3 almost surely.

Proof. Let E be the image of a reflected Brownian excursion under the map b−1θ

(z) = z5/8 fromH to

the wedge W (3π/8). It is easy to verify that the right boundary of E [0,∞) is b−1θ

(∂r (bθ(E [0,∞)))).

Let γ(t ) be a parametrisation of ∂r E [0,∞), which has the same law as b−1θ

γ where γ is an SLE8/3

by Theorems 3.3.3 and 3.5.3. Also, b−1θ

preserves Hausdorff dimension by Propositions 3.6.3 and

3.6.4. Therefore, dimH ∂r E [0,∞) = 4/3 almost surely.

Define the stopping times σ= inft ≥ 0 : Re Et = 2, τ= inft ≥σ : Re Et = 1. By Proposition 2.1.8,


θ

0

τr

σ1

σ2

Figure 7: The labels σ1, σ2, and τ are displayed near the points Eσ1 , Eσ2 , and Eτfor a linear interpolation of a sample path E [0,τ]

σ and τ are finite almost surely. Let σ1 and σ2 be random times defined by

σ1 = infσ≤ t ≤ τ : Et = supIm Es : Re Es = 2 and σ≤ s ≤ τ

σ2 = infσ≤ t ≤ τ : Et = infIm Es : Re Es = 2 and σ≤ s ≤ τ

Let r = inf |Eτ− Et | : t is between σ1 and σ2, and let τ′ be the neighbourhood rational randomtime associated with τ and r . (Note that r > 0 by compactness). By the continuity of E ,

∂r,1E [σ,τ] = ∂r,1E [σ,τ′]. (3.8.1)

θ

(1,n)

∆(Eτ,n−1)

Figure 8: The shaded region represents the domain Un(z,θ).

Proposition 2.1.9 shows that dimH ∂r,2E [σ,τ] ≤ d intersects (g γ)(I ) on an interval I of positivelength, which implies dimH ∂r,2E [σ,τ] ≥ 4/3 almost surely by Proposition 3.6.7. Suppose that theevent C B dimH ∂r,2E [σ,τ] > 4/3 occurs with positive probability. Let ∆(z,r ) denote the opendisk of radius r centred at z. For z with real part 1 and n ∈ N, define the domain Un(z,θ) (seeFigure 8) by

Un(z,θ) =∆(Eτ,1/n)∪ (W (θ) \ ([1,∞)× [0,n])) .

Define the events Dn = E [τ,∞) ⊂ Un(Eτ,θ)). By Propositions 2.5.3 and 2.2.6, P(Dn) > 0 for alln ∈N. By the monotone convergence theorem, P(C∩E(σ,τ)∩Un(Eσ,θ) =;) > 0 for large enough


n. By the strong Markov property, C and Dn are independent. Therefore, P(C ∩Dn) > 0. Moreover,we may cover ∂r,2E(σ,τ] with countably many open disks which do not intersect Eσ. If ∂r,2E(σ,τ]has dimension greater than 4/3, then its intersection with at least one of these disks has dimensiongreater than 4/3 as well, by Proposition 3.6.4. Index the disks by N and define ∆ to be the diskwith least index whose intersection with ∂r,2E [σ,τ] has dimension greater than 4/3. With positiveprobability, E [0,σ] does not intersect ∆. Since E [0,σ] is independent of (Eσ+t − Eσ)t≥0, we findthat ∆∩ ∂r,2E [σ,τ] coincides with (b−1

θγ)[0,∞) on an interval of positive length, with positive

probability. Again by Proposition 3.6.7, this is a contradiction. Therefore, dimH ∂r,1E [σ,τ] = 4/3almost surely. From (3.8.1), we also have dimH ∂r,1E [σ,τ′] = 4/3 almost surely.

Since E [σ,τ′] has the same law as a Brownian excursion started at Eσ, up to time τ′, we concludeby Lemma 3.8.7 (along with a suitable translation) that dimH ∂r,0E [0,T ] = 4/3 almost surely for aBrownian excursion E started at i and a stopping time T whose range is countable. Let T denotethe neighbourhood rational random stopping time associated with the hitting time of R+3i /2 andradius r = 1/2, and let T ′ be the neighbourhood rational random time associated with the hittingtime of R+ i /2 and radius r = 1/2. By Proposition 3.2.3, E [0,T ∧T ′] has the same distribution asB[0,T ∧T ′], where B is a Brownian motion started at i and conditioned to hit R+2i before R. ThusdimH ∂r,0B [0,T ∧T ′] = 4/3 almost surely. Since B is a conditioned Brownian motion, we have

1 =P(dimH ∂r,0B [0,T ∧T ′] = 4/3

)= 2P(

dimH ∂r,0B [0,T ∧T ′] = 4/3 and B hits R+2i before R)

,

which by symmetry implies P(dimH ∂r,0B [0,T ∧T ′] = 4/3) = 1. By Lemma 3.8.7, dimH ∂r,0B [0,1] =4/3 almost surely. By Proposition 3.8.6 and symmetry, dimH frB [0,1] = 4/3 almost surely.

REFERENCES 27

REFERENCES

[1] Vincent Beffara. The dimension of the SLE curves. The Annals of Probability, 36(4):1421–1452,2008.

[2] Nathanaël Berestycki. Stochastic calculus and applications. http://www.statslab.cam.ac.uk/∼beresty/teach/StoCal/sc3.pdf.

[3] John Cardy. SLE for theoretical physicists, 2005. http://arxiv.org/abs/cond-mat/0503313v2.

[4] Gregory Lawler. The dimension of the frontier of planar Brownian motion. Electron. Comm.Prob., 1(5):29–47, 1996.

[5] Gregory Lawler. Conformally Invariant Processes in the Plane. American Mathematical Soci-ety, 2005.

[6] Gregory Lawler, Oded Schramm, and Wendelin Werner. The Hausdorff dimension of the pla-nar brownian frontier is 4/3, 2000. http://arxiv.org/abs/math/0010165v2.

[7] Gregory Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordalcase. Journal of the American Mathematical Society, 16(4):917–955, 2003.

[8] James Norris. Introduction to Schramm-Loewner evolutions. http://www.statslab.cam.ac.uk/∼james/Lectures/sle.pdf.

[9] Bruce Palka. An Introduction to Complex Function Theory. Springer, 1991.

[10] Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 3rdedition, 1999.

[11] L.C.G. Rogers and David Williams. Diffusions, Markov Processes and Martingales. CambridgeUniversity Press, 1994.

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