Brownian Motion, Complex Analysis, and the Dimension of the Brownian Frontier

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 the Brownian frontier

Sam Watson Trinity 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 and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualiﬁcation at any university or similar institution.

Signed:

58 County Road 362 Oxford, Mississippi 38655 USA Brownian motion, complex analysis, and the dimension of the Brownian frontier Contents

1 Introduction 1

2 Brownian Motion and Complex Analysis1 2.1 One-dimensional Brownian motion...... 1 2.2 Brownian motion in Rd ...... 4 2.3 Conformal invariance of planar Brownian motion ...... 7 2.4 Applications to complex analysis...... 8 2.5 Applications to planar Brownian motion...... 9

3 Schramm-Loewner evolutions and the dimension of the Brownian frontier 11 3.1 Overview...... 11 3.2 Brownian excursions...... 11 3.3 Reﬂected Brownian excursions...... 13 3.4 Loewner’s theorem...... 15 3.5 Restriction property of SLE8/3 ...... 17 3.6 Hausdorff dimension ...... 18 3.7 Hausdorff dimension of SLE ...... 20 3.8 Hausdorff dimension of the Brownian frontier...... 21

iv 1 INTRODUCTION

This essay explores the interplay between complex analysis and planar Brownian motion. In Sec- tion2, we develop basic properties of Brownian motion and give probabilistic proofs of classical theorems from complex analysis. We discuss conformal invariance and illustrate how the theory of conformal maps in can be used to prove statements about planar Brownian motion. Section3 culminates in a proof that the frontier of planar Brownian motion has Hausdorff dimension 4/3 almost surely. This proof is made possible by a recent development called the Schramm-Loewner evolution, 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 analysis drawn mainly from Rogers and Williams’sbook [11] and Nathanaël Berestycki’snotes on Stochas- tic Calculus [2]. For the review of Schramm-Loewner evolutions and Brownian excursions, James Norris’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 is from 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 planar 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 defined ≥ on a probability space (Ω,F,P) which satisfies the following conditions. (i) B 0 almost surely, 0 = (ii) for almost all ω Ω, t B (ω) is continuous, and ∈ 7→ t (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 is ≥ ≥ an (Ft )t 0-Brownian motion if the following conditions hold. ≥ (i) B 0 almost surely, 0 = (ii) for almost all ω Ω, t B (ω) is continuous, ∈ 7→ t (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 is − ≥ ≥ a Brownian motion started at X . See [2] for a proof of the following theorem, which conﬁrms the existence of Brownian motion.

Theorem 2.1.3. (Wiener) There exists a process (Bt )t 0 satisfying conditions (i)-(iii) in Deﬁnition ≥ 2.1.1.

© 1 2 3 ª Idea of proof. Deﬁne D 0, n , n , n ,...,1 , and let {Z : n Z+,d D } be a countable collec- n = 2 2 2 n,t ∈ ∈ n tion of iid standard normal random variables. Set B (0) 0, B (0) Z , and B (0) t Z (i.e., linearly 0 = 1 = 0,1 t = 0,1 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 2

(n 1) (n) (n 1) interpolated between the values at 0 and 1). Define Bt − inductively so that Bt Bt − for n n = t Dn 1. For t Dn \ Dn 1, set t − t 2− , t + t 2− , and ∈ − ∈ − = − = + (n 1) (n 1) B − B − (n 1) t − t Zn,t B + + + t = 2 + 2n In other words, we have added a Gaussian fluctuation with the appropriate variance at the new (n) dyadic times. Again, extend Bt to [0,1] by linear interpolation. Using some elementary estimates, (n) one can show that Bt is uniformly Cauchy, from which it follows that there exists a uniform limit ³ (n)´ Bt . This limit inherits independent, normally distributed intervals from Bt . We obtain a t Dn Brownian motion on [0, ) by summing a countable collection of independent copies∈ of Brownian ∞ motion defined on unit intervals.

An adapted process (Xt )t 0 for which (Xt , Xt ,..., Xt ) is jointly Gaussian for all finite sets of times ≥ 1 2 n (tk )1 k n is called a Gaussian process. If EXt 0, then (Xt )t 0 is called a mean-zero process. It ≤ ≤ = ≥ is 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 X s t for all s,t 0. t s = ∧ ≥ 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 ∞ X∞ n d(f ,g) 2− sup f (x) g(x) . = n 1 0 x n | − | = ≤ ≤ We equip W with its Borel σ-algebra W B(W ). (We use the notation B(X ) for the Borel σ-algebra = on a topological space X .)

We may think of a Brownian motion B as a map Ω W which sends ω to the continuous function → t B (ω). The image measure of B is called the Wiener measure and is denoted W. 7→ t 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 cB 2 is a Brownian motion (scaling). > t/c (iii) The process starting at 0 and equal to tB for t 0 is a Brownian motion (time inversion). 1/t > Proof. The processes (i), (ii), and (iii) are mean-zero Gaussian processes, since they inherit jointly Gaussian ﬁnite-dimensional distributions from (Bt )t 0. To check the covariance, we compute ≥ E( B , B ) E(B ,B ) s t for (i), − s − t = s t = ∧ £ ¤ 2 £ ¤ 2 2 2 E (cB 2 )(cB 2 ) c E B 2 B 2 c (s/c t/c ) s t s/c t/c = s/c t/c = ∧ = ∧ for (ii), and

E[(sB )(ctB )] st E[B B ] st(1/s 1/t) min(s,t) 1/s 1/t = 1/s 1/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 same > ∞ distribution as C((0, ),R)-valued random variables. Thus ∞ limsup Bt 0, a.s. limsup tB1/t 0, a.s. t 0 | | = ⇒ t 0 | | = → → 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 time ≤ ∈ ≥ ≥ T , the σ-algebra FT is defined by F {A F : A {T t} F for all t 0}. T = ∈ ∩ ≤ ∈ t ≥ 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 3

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 Markov property or the simple Markov property. We give one consequence of the Markov property which we will need later. Proposition 2.1.8. With probability 1, limsup B and limsup B . t t = ∞ t t = −∞ Proof. (From [11]). Let X sup B . By scaling, cX has the same distribution as X for all c 0. = t 0 t > 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) = + ≤ ∀ ≥ 1 p, = 2 which implies p 0. = Proposition 2.1.9. For all ² 0, the set {B : 0 t ²} almost surely contains both positive and > t ≤ ≤ negative numbers.

Proof. Apply Proposition 2.1.5(iii) to Proposition 2.1.8. Since Bt almost surely takes on arbitrarily 1 large and small values as t ranges over the interval (²− , ), we ﬁnd that (tB1/t )t 0 almost surely ∞ > takes on positive and negative values as t ranges over the interval (0,²).

We record the following important result about martingales. For a proof, see [11]. Recall that a martingale (Mt )t 0 is deﬁned to be an adapted process for which E Mt for all t 0 and ≥ | | < ∞ ≥ E(M M F ) 0, 0 s t . t − s| s = ∀ ≤ ≤ < ∞ Also, recall that a collection X of random variables is said to be uniformly integrable if

supE( X 1 X K ) 0 as K . X X | | | |> → → ∞ ∈ Theorem 2.1.10. (Optional Stopping Theorem) Let (Mt )t 0 be a continuous adapted process for ≥ which E M for all t 0. Then the following are equivalent. | t | < ∞ ≥ (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) EX EX for all bounded stopping times T . 0 = T Also, if (Mt )t 0 is a uniformly integrable martingale, then E(XT FS) XS T for all stopping time ≥ | = ∧ S,T .

The following proposition allows us to give bounds for the probability that sup{ B : 0 t ²} x. | t | ≤ ≤ < 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 . = ≤ ≤ ≥ | | 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 4

Proof. For all t 0 and a 0, we have ≥ ≥ P(S a) P(S a and B a) P(S a and B a) t ≥ = t ≥ t ≤ + t ≥ t > P(B a) P(B a) = t ≥ + t > 2P(B a) P( B a). = t ≥ = | t | ≥ The step P(S a and B a) P(B a) comes from the reflection principle [2], which asserts t ≥ t ≤ = t ≥ that for all a,b R, P(S a and B b) P(B 2a b). ∈ t ≥ t ≤ = t ≥ − We recall a few basic facts from stochastic calculus which are needed for the proof of the following proposition. Proofs of these statements may be found in [2]. A real-valued process (Mt )t 0 is said ≥ to be a local martingale if there exists a sequence (Tn)n 1 of stopping times tending to as n ≥ ∞ → ∞ for which (MT t )t 0 is a martingale for all n. For every continuous local martingale, there exists a n ∧ ≥ unique continuous adapted nondecreasing process, denoted [M]t and called the quadratic varia- 2 tion of (M)t 0, for which (M [M]t )t 0 is a continuous local martingale. A bounded continuous ≥ − ≥ local martingale is a true martingale, and the quadratic variation of a Brownian motion (Bt )t 0 is ≥ t. Proposition 2.1.12. Let 0 a b, let B be a Brownian motion started at a, and let τ inf{t 0 : < < r = ≥ B r }. Then P(τ τ ) a/b, and E(τ τ ) a(b a). t = 0 > b = 0 ∧ b = − Proof. Let p P(τ τ ), and apply the optional stopping theorem to the bounded martingale = 0 > b (Bt )t τ0 τb to find that a EM0 EMτ0 τb pb (1 p)0. For the second equality, note that 2 ∧ ∧ = = ∧ = 2+ − 2 2 B t τ0 τb is a martingale. Therefore, EB Et τ0 τb EB a . Take t and t τ0 τb − ∧ ∧ t τ0 τb − ∧ ∧ = 0 = → ∞ apply∧ ∧ dominated convergence for the first term and∧ monotone∧ convergence for the second term to find

b2(a/b) 0 Eτ τ a2, + − 0 ∧ b = which gives Eτ τ a(b a). 0 ∧ b = −

2.2 Brownian motion in Rd

Deﬁnition 2.2.1. A Brownian motion in Rd is a d-dimensional vector whose components are independent 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 stochastic calculus. For a proof, see [2]. Theorem 2.2.2. (Lévy’s characterisation) Let M (M 1,...,M d ) be an Rd -valued random process = whose components are continuous local martingales. Then M is a Brownian motion if and only if ½ t if i j [M i ,M j ] = t = 0 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 UB is a Brownian motion.

Proof. Let u1,u2,...,ud be the rows of the matrix U. As a linear combination of continuous local martingales, the components of UB are continuous local martingales. Also, the ith component of UB is (UB) uT B, so [(UB) ,(UB) ] uT u t, which is t if i j and is 0 otherwise, by the i = i i j t = i j = deﬁnition of orthogonality. By Lévy’s charactisation, UB is a Brownian motion. 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 5

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 we ≥ will 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 and ∞ × → twice continuously differentiable in the second coordinate. Suppose that there exists K 0 for > which ¯ ¯ d ¯ ¯ d ¯ 2 ¯ ¯∂f ¯ X ¯ ∂f ¯ X ¯ ∂ f ¯ f (t,x) ¯ (t,x)¯ ¯ (t,x)¯ ¯ (t,x)¯ K exp(K (t x ), (2.2.1) | | + ¯ ∂t ¯ + i 1 ¯∂xi ¯ + i,j 1 ¯∂xi ∂x j ¯ ≤ + | | = = for all (t,x) [0, ) Rd . Then ∈ ∞ × Z t µ ∂ 1 ¶ Ct B f (t,Bt ) f (0,B0) f (s,Bs)ds − − 0 ∂t + 24 is a martingale. Here denotes the Laplacian Pd ∂2/∂x2. In particular, if f is harmonic (that is, 4 i 1 i f is identically zero), then f (t,B ) is a martingale.= 4 t 2 d/2 Proof. First we will show that E(C C ) 0 for 0 s t. Denote by p (x) exp( x /2t)/(2πt)− t − s = < ≤ t = −| | the density of Bt . We compute µ Z t µ ∂ 1 ¶ ¶ E(Ct Cs) E f (t,Bt ) f (s,Bs) f (u,Bu)du − = − − s ∂t + 24 Z Z t Z µ ¶ ¡ ¢ ∂ 1 pt (x)f (t,x) ps(x)f (s,x) dx pu(x) f (u,x)dx du = Rd − − s Rd ∂t + 24 Z Z t Z ¡ ¢ ∂ pt (x)f (t,x) ps(x)f (s,x) dx (pu(x)f (u,x))dx du, = Rd − − s Rd ∂u where we have used integration by parts twice to rewrite R p (x) f (u,x)dx as R p (x)f (u,x)dx. u 4 4 u Here we require the growth condition (2.2.1) so that the product pu(x)f (u,x) decays sufﬁciently rapidly as x that the boundary terms in the integration by parts are zero. We have also used | | → ∞ 1 p (x) ∂p (x)/∂u. Now we may interchange the integrals in the second term, since the mod- 2 4 u = u ulus of the integrand is bounded. An appeal to the fundamental theorem of calculus completes the proof that E(C C ) 0. t − s = Now we will to take s 0 to show that E(C ) 0. First, observe that C C C pointwise as s 0. → t = t − s → t → By Propositions 2.1.11 and 2.1.5(ii), we have E(sup0 u t Bu a) 2E( B1 a/ pt). Thus ≤ ≤ | | ≥ ≤ | | ≥ ¯ Z t µ ¶ ¯ ¯ ∂ 1 ¯ Ct Cs ¯f (t,Bt ) f (s,Bs) f (u,Bu)du¯ | − | = ¯ − − s ∂u + 24 ¯ Z t ¯µ ¶ ¯ ¯ ∂ 1 ¯ f (t,Bt ) f (s,Bs) ¯ f (u,Bu)¯ du ≤ | | + | | + s ¯ ∂u + 24 ¯ µ µ ¶¶ (2 t)K exp K t sup Bu , ≤ + + 0 u t | | ≤ ≤ which is integrable because µ ¶ µ µ ¶ ¶ X∞ Eexp K sup Bu 1 P exp K sup Bu n 0 u t | | ≤ + n 1 0 u t | | ≥ ≤ ≤ = µ ≤ ≤ ¶ X∞ logn 1 2 P X1 ≤ + n 1 | | ≥ K pt = X∞ K pt 1 4 exp( (logn)2/2K 2t) . ≤ + n 1 p2πlogn − < ∞ = 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 6

1 We have used the estimate E(X x) x− exp( x2/2) for the tail of the normal distribution. Thus 1 ≥ ≤ 2π − E(C ) 0 for all t. By the Markov property of Brownian motion, this gives E(C C F ) 0, as t = t − s| s = desired.

Theorem 2.2.5. Planar Brownian motion is neighbourhood-recurrent but does not visit a pre- speciﬁed point. More precisely, let B be a planar Brownian motion started at z C. Then for t 0 ∈ all open U C, the set ⊂ {t 0 : B U} (2.2.2) ≥ t ∈ is almost surely unbounded, and for all w , z0, {t 0 : B w} , (2.2.3) ≥ t = = ; 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 log z by a smooth function which is equal to 1 on {z : a z b} 7→ | | ≤ | | ≤ and 0 on {z : a/2 z 2b} to obtain a smooth, compactly supported function f which satisfies ≤ | | ≤ f (z) log z for all a z b. Then f trivially satisfies (2.2.1). Therefore, f (Bt ) is a martingale. = | ∂|2 f ∂2 f | 0| 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 at d ≥ d 2 z R never visits the closed ball of radius 0 r z centered at the origin is (r / z ) − . ∈ < < | | | |

Proof. Let Bt be a d-dimensional Brownian motion, and let τr denote the hitting time of the closed 2 d d ball of radius r centered at the origin. Since the function z z − is harmonic in R \{0}, we ﬁnd 2 d 7→ | | using Theorem 2.2.4 that Mt B Bt τ − is a martingale (after smoothing the singularity at the | ∧ r | origin as in the proof of Theorem 2.2.2). 2 d Let s z , let τ be the hitting time of {z : z s}, and let φ(x) x − . The optional stopping > | | s | | = = theorem applied to Mt implies

φ(x) EM0 EMτ τ P(τr τs)φ(r ) P(τr τs)φ(s), = = r ∧ s = < + ≥ φ(s) φ( z ) d 2 which gives P(τr τs) − | | . Taking s , we ﬁnd P(τr ) (r / z ) − . < = φ(s) φ(r ) → ∞ < ∞ = | | − 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 7

2.3 Conformal invariance of planar Brownian motion

A connected open subset of C is called a domain. Let D be a domain, let f : D C be a nonconstant → analytic function, and let (BT t )t 0 be a Brownian motion started at z D and stopped at the exit ∧ ≥ ∈ time T inf{t 0 : B C \ D}. By the open mapping theorem [9], f (D) is an open set. Therefore, = ≥ t ∈ if we consider the image of (BT t )t 0 under f , we obtain a C-valued stochastic process started at ∧ ≥ f (z) in the domain f (D) and stopped when it hits C \ f (D) (see Figure1).

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

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

For all z0 D with f 0(z0) , 0, the Taylor series development ∈ 2 f (z) z f 0(z )(z z ) O( z z ), = 0 + 0 − 0 + | − 0| shows that f transforms points near z by scaling z z by a factor of f 0(z ) and rotating z z 0 − 0 | 0 | − 0 by an angle arg f 0(z0). Since Brownian motion is invariant under rotations and scaling (the latter with a time change), we expect that f (Bt )t 0 can be transformed into a Brownian motion with a ≥ suitable time change. (The restriction that f 0(z0) , 0 does not pose a problem since the zeros of f are isolated). This property of Brownian motion is called conformal invariance, in reference to the case where f is a conformal map (that is, an injective analytic function). In preparation for a proof, we recall two theorems from stochastic calculus. Proofs may be found in [2]. d 1 Theorem 2.3.1. (Itô’s formula) Let D be a domain in R + , 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 exit 1 d time of (t,Mt ,...,Mt ) from D, we have Z t 1 d 1 d ∂f 1 d f (t,Mt ,...,Mt ) f (0,M0 ,...,M0 ) (s,Ms ,...,Ms )ds = + 0 ∂t d Z t X ∂f 1 d i (s,Ms ,...,Ms )dMs + i 1 0 ∂xi = d d Z t 2 1 X X ∂ f 1 d i j (s,Ms ,...,Ms )d[M ,M ], + 2 i 1 j 1 0 ∂xi ∂x j = = i j i j i j i j where [M ,M ]t B ([M M ]t [M M ]t )/4 denotes the covariation of M and M . + − − Theorem 2.3.2. (Dubins-Schwarz) Let M be a continuous local martingale for which M 0 al- 0 = most surely and limt [M]t almost surely. Let σ(t) inf{s :[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 adapted ≥ ≥ to (Fσ(t))t 0. ≥ 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 8

Theorem 2.3.3. (Conformal invariance of Brownian motion) Let D be a domain and let (Bt )t 0 be ≥ a Brownian motion started at z D. If f : D C is analytic, then there exists a Brownian motion ˜ ∈ ˜→ Bt in f (D) started at f (z) for which f (Bt ) BR t f (B ) 2 ds. = 0 | 0 s | Proof. Write z x i y, f (z) u(x, y) i v(x, y), and B X iY where X and Y are independent = + = + t = t + t t t scalar Brownian motions. Recall that the real and imaginary parts u and v of an analytic function f u i v satisfy the Cauchy-Riemann equations = + ∂u ∂v ∂u ∂v , . ∂x = ∂y ∂y = −∂x By taking further partial derivatives, we see from these equations that u and v are harmonic. Therefore, Itô’s formula gives ∂u ∂u du(Xt ,Yt ) (Xt ,Yt )d Xt (Xt ,Yt )dYt = ∂x + ∂y 1 µ∂2u ∂2 y ¶ 1 ∂2u dt (Xt ,Yt )d[X ,Y ]t + 2 ∂x2 + ∂y2 + 2 ∂x∂y ∂u ∂u (Xt ,Yt )d Xt (Xt ,Yt )dYt . = ∂x + ∂y

Similarly, dv(Xt ,Yt ) ∂v/∂x(Xt ,Yt )d Xt ∂v/∂y(Xt ,Yt )dYt . From these expressions we may com- = + R t 2 pute the quadratic variation [u(B)]t [v(B)]t 0 f 0(Bs) ds and the covariation [u(B),v(B)]t 0. R s 2 = = | | = Let σ(t) inf{s 0 : f 0(B ) du t}, and deﬁne B˜ f (B ) u(B ) i v(B ). By the = ≥ 0 | u | > t = σ(t) = σ(t) + σ(t) Dubins-Schwarz theorem, B˜t 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 fundamental results in complex analysis. We begin with a lemma stating a basic property of harmonic functions in preparation for a proof of the maximum modulus principle. Proofs of Theorem 2.4.2 and 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 = ∈ | − | < < Z 2π h(z0) h(z r exp(iθ))dθ. = 0 +

Proof. Let τ be the exit time of a planar Brownian motion started at z0 from the disk of radius r centred at z0. By Theorem 2.2.4,(h(Bt ))t 0 is a martingale. Apply the optional stopping theorem ≥ to ﬁnd that h(z ) Ph(B ) R 2π h(z r exp(iθ))dθ, as desired. 0 = τ = 0 + 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 z U and ² 0 for which f (z ) 0 ∈ > | 0 | ≥ f (z r exp(iθ) for all 0 r ². By adding a suitable constant to f , we may assume that the | 0 + | < < image of {z : z z ²} under f is contained in the right half-plane. Therefore, log f is analytic | − 0| ≤ on {z : z z ²}, from which it follows that the real part log f is harmonic. By the previous | − 0| < | | theorem, for any 0 r ² we have that log f (z ) is an average of the values log f (z r exp(iθ) . < < | 0 | | 0 + | Therefore, log f is constant on ∆ B {z : z z0 ²}. This implies that f is constant on ∆, which | | | − | < | | in turn implies that f is constant on ∆ by the Cauchy-Riemann equations. Since U is connected, f is constant on U. 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 9

Theorem 2.4.3. (Fundamental Theorem of Algebra) If p is a nonconstant polynomial, then there exists 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 since ∈ = p as z , f is bounded. Let B be a Brownian motion started at the origin. By Theorem → ∞ → ∞ t 2.2.4, Re f (Bt ) is a martingale. Since Re f is bounded, the martingale convergence theorem implies that Re f (B ) converges almost surely as t . On the other hand, Re f (C) contains more than t → ∞ one element, since f is nonconstant. Choose α β so that infRe f (C) α β supRe f (C). The < < < < sets U1 B {z : Re f (z) α} and U2 B {z : Re f (z) β,} are nonempty, disjoint open sets in C. By < > Theorem 2.2.5, the Brownian motion visits each of the sets U1 and U2 at arbitrarily large times, so liminfRe f (B ) α β limsupRe f (B ), almost surely. This contradicts the convergence of t < < < t Re f (Bt ). Theorem 2.4.4. (Schwarz Lemma) Let D {z : z 1} denote the unit disk. If f : D D is an = | | < → analytic 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) eiθz for all z D. | | = | | ∈ = ∈ Proof. Let z D and choose 0 r 1 so that z is contained in the open disk centred at the origin 0 ∈ < < 0 with radius r . Let S denote the hitting time of the circles of radius r . The function g(z) f (z)/z is = continuous at the origin since limz 0 g(z) f 0(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 optional stopping theorem to g(Bt ) to ﬁnd

g(z ) Eg(B ) 0 = S Since f (z) 1 for all z D, we have g(B ) f (B ) / B 1/r . Letting r 1 gives g(z ) 1. | | ≤ ∈ | S | = | S | | S| ≤ → | 0 | ≤ Moreover, if g(z ) 1, then z is a local maximum for g. By the previous theorem, this implies | 0 | = 0 that 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 Brownian motion. 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 a time change of a Brownian motion. Since Brownian motion visits every neighbourhood of , the R 2 ∞ boundedness of f requires that that the time change does not go to . That is, ∞ f 0(B ) ds ∞ 0 | s | < ∞ almost surely. We claim that this holds only if f is constant. For if f is nonconstant, then we may 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 S and T to be the nth entrance and > ≥ ∈ n n exit times, respectively, of D. Then Sn and Tn are finite for all n almost surely. By the Markov property of Brownian motion, the variables Tn Sn are i.i.d. Also, each has finite expectation by − R 2 P 2 Proposition 2.1.12. By the strong law of large numbers, ∞ f 0(B ) dx ∞ δ (T S ) 0 | s | ≥ n 1 n − n = ∞ almost surely. =

2.5 Applications to planar Brownian motion

Conformal invariance and properties of analytic maps can also be used to answer questions about planar Brownian motion. The latter half of the paper is devoted to carrying out this task for the 2 BROWNIAN MOTION AND COMPLEX ANALYSIS 10 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 (²,0). Denote by n² the number of windings of Bt around the origin before the first time Bt hits the unit circle centred at the origin. For all 0 ² 1, the law of 2πn /log² is the Cauchy distribution. < < ²

Proof. Let Xt be a Brownian motion started at (log²,0) and stopped at τ inf{t 0 : Re Xt 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 Xt is equal in distribution to 2πn², since Im Xt is a continuous realisation of the multi-valued function argexp Xt which starts at 0 (see Figure2).

(log²,0) z exp(z) 7→ (²,0) 0 0

Xt Bt

Figure 2: The Brownian motion Xt started at (log²,0) and its image under the exponential map.

Define x log² and S sup (Re X x), and observe that τ t if and only if S x. Com- = − t = 0 s t s + < t < bining Propositions 2.1.5(ii) and≤ 2.1.11≤ , we find that the random time τ has the same distribution as x2/Z 2, where Z a standard normal random variable. Also, because τ and Im X are independent, d we may apply the scaling property of Brownian motion to obtain Im X τ1/2 Im X (to see that τ = 1 this equality holds for all stopping times independent of X , note that it is straightforward for a ³ ´ 1 d ¡ x ¢ Im X1 stopping times of the form n− τ/n , and take n ). Thus Im X (Im X ) x . Re- d e → ∞ τ = Z 1 = Z calling that the quotient of two independent standard normal random variables has the Cauchy d d distribution, we have Im Xτ C x, where C is a Cauchy random variable. Therefore, 2πn² C x d = = ⇒ 2πn /log² C. ² = Given a subset S of the boundary of a domain D, one expects that (under suitable regularity conditions on D), a Brownian motion started at a point in D has positive probability of exiting D at a point in S. We will state and prove this result in the case that D is a Jordan domain. Recall that a Jordan domain is defined to be a simply connected domain in C whose boundary is the trajectory of 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 let τ denote the hitting time of C \ D. If S ∂D has positive arc length, then P(B S) 0. ⊂ τ ∈ > Proof. Since D is a Jordan domain, there exists a conformal map ϕ from D to the unit disk which sends z to 0 and which extends to a homeomorphism from the closure of D to the closed unit disk [9]. Let B 0 be a Brownian motion in the unit disk with τ0 the hitting time of the unit circle. By con- ¯ formal invariance, P(Bτ S) P(B 0 ϕ(S)). Since ϕ¯ : ∂D {z : z 1} is a homeomorphism, ∈ = τ0 ∈ ∂D → | | = ϕ(S) has positive arc length. By rotational symmetry, P(B 0 ϕ(S)) (the length of S)/(2π) 0. τ0 ∈ = > 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 of Rn, then the Hausdorff dimension is (roughly) the exponent d for which the number of balls of radius r needed to cover A scales as 1/r d as r 0. For a path in R2 whose → trajectory exhibits similar behaviour when viewed on arbitrarily small scales, we may think of the Hausdorff 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 Figure5 have dimension strictly between 1 and 2, as we shall see. If B is a Brownian motion in Rn for n 2, then B[0,1] has ≥ Hausdorff 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 of → C\γ[0,t]. Informally, it is “the trajectory γ[0,t] with the holes filled in.” The boundary of the hull is called 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 that the Hausdorff dimension of the Brownian frontier is 4/3. This conjecture was proved in 2000 by Lawler, Schramm, and Werner [6] using the Schramm-Loewner evolution SLEκ, which is a family of 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 intersecting exponent ξ(2,0), which is defined by ¡ ¢ ξ(2,0) o(1) P B[0,T ] B 0[0,T 0 ] does not disconnect the unit circle from R− + , R ∪ R ∞ = where B and B 0 are independent Brownian motions and TR and TR0 are their exit times from the disk of radius R. They show that the intersecting exponents are the same for any two random paths whose laws are completely conformally invariant. They conclude the proof by computing the 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 of Beffara [1]. The proof uses a calculation of the Hausdorff dimension of the SLE paths [1] along with work of Lawler, Schramm, and Werner which relates the Brownian frontier to SLE8/3 [7]. We begin with a discussion of Brownian excursions, which prove to be an indispensible tool for making the connection 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 started at 0. Deﬁne E X i W .A Brownian excursion from 0 to in H is a process which is equal in t = t + | t | ∞ distribution to (Et )t 0. A Brownian excursion can be viewed as a Brownian motion conditioned so ≥ that its imaginary part remains positive at all positive times, as we will see in Proposition 3.2.3. In preparation for the proof of this theorem, we recall two theorems from stochastic calculus. Proofs may be found in [2] Theorem 3.2.1. (Girsanov) Let M be a continuous local martingale on a probability space (Ω,F,P), 1 and suppose that the exponential martingale Zt B exp(Mt [M]t ) of M is uniformly integrable − 2 and that M0 0 almost surely. Deﬁne a measure P˜ by dP˜ /dP Z . If X is a continuous local = = ∞ martingale 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 stochastic → → differential equation

dX σ(X )dB b(σ )dt, (3.2.1) t = t t + t 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 12 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 (Xt iYt )t 0 = + ∈ > ≥ = + ≥ started at z and conditioned to hit R iR before R is equal in distribution to (Wt i W˜ t )t 0, where + + | | ≥ W and W˜ are independent standard Brownian motions started at 0 in R and at (y,0,0) in R3, respectively.

1 Proof. For r 0, let τr denote the hitting time of R ir . Define Mt y− Yτ0 τR t , and observe ≥ + =R ∧ ∧ that Mt is a bounded nonnegative martingale with M limt Mt 1{τ τ }. We define a new ∞ = →∞ = y 0> r measure P˜ by dP˜ /dP M , where P is the law of (Bt )t 0. Under P˜ , we have τ0 τR almost surely. = ∞ ≥ R t >1 To apply Girsanov’s theorem, note that Mt is the exponential martingale of 0 Mt− dMt . Therefore, the processes (Xt )t 0 and ≥ Z t 1 Bt Yt Ms− d[Y ,M]s = − 0 are continuous local martingales under P˜ . Since [X , X ] [B,B] t and [X ,B] 0, X and B are t = t = = independent P-Brownian motions by Lévy’s characterisation. Moreover, under P˜ , Z t ds Yt y Bt . = + + 0 Ys Using Itô’s formula with f (x) x in the domain R3 \ {0}, we find = | | 3 Z t ˜ i Z t X Ws 1 2ds W˜ W˜ dW˜ i t o ˜ s ˜ | | = | | + i 1 0 Ws + 2 0 Wt = | | | | The first term on the right-hand side is y and the second term is a standard Brownian motion, by Lévy’s characterisation. Therefore, the process ( W˜ t )t 0 satisfies the same stochastic differential | | ≥ equation (3.2.1) as Y under P˜ , with σ 1 and b(x) 1/x. Although b is not Lipschitz, we may t = = replace b with a Lipschitz function bn which is equal to b up to time τ1/n. By Theorem 3.2.2, this implies that for all n,(Yt )t 0 and ( W˜ t )t 0 have the same law up to time τ1/n. Taking n , we ≥ | | ≥ → ∞ find that (Yt )t 0 and ( W˜ t )t 0 have the same law up to time τ0. Moreover, the law of Yt under P ≥ | | ≥ conditioned on the event τ τ is the same as the law of Y under P˜ , because Proposition 2.1.12 R < 0 t gives

R P(Zt A and τ0 τR ) P˜ (Zt A) E(M 1{Z A}) P(Zt A and τ0 τR ) ∈ > , ∈ = ∞ t ∈ = y ∈ > = P(τ τ ) 0 > R for all A B(C([0, ),C). The last quotient is the elementary conditional probability of A given ∈ ∞ that Z hits R iR before R. t + Definition 3.2.4. A bounded set K H for which K H K and H\K is simply connected is called ⊂ ∩ = a compact H-hull. The set of compact H-hulls is denoted Q. The set Q is defined to contain the + compact H-hulls whose intersection with the negative ray ( ,0] is empty. The set Q is defined −∞ − analogously 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 conformal maps, 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 map g : H \ K H which satisfies g (z) z 0 as z . K → K − → | | → ∞ The next theorem provides a further connection between compact H-hulls K and their associated 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 13

maps gK . We use the notation X [a,b) for the trajectory {Xt : a t b} of a process (Xt )t 0 on the ≤ < ≥ time 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 0 (0). ∞ ∩ = ; = K

Proof. First observe that by the Schwarz reﬂection principle, gK can be extended to an analytic function 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, gK0 (0) exists. By the continuity of g on N H, we have g(N R) R. This implies that g 0 (0) is real, which in turn implies g 0 (0) ∩ ∩ ⊂ K K = limw 0 ImgK (w)/Imw. → Now let τ be the hitting time of R iR. Let t 0 and let B be a Brownian motion started at E . By R + > t the previous proposition, P(B hits R iR before R K ) P(E[t,τR ) K ) + ∪ . (3.2.2) ∩ = ; = P(B hits R iR before R) + A Brownian motion B started at z H has probability Im(z)/R of hitting R iR before R, by Proposi- ∈ + tion 2.1.12. The probability that B hits R iR before R K is equal to the probability that a Brownian + ∪ motion started at g (z) hits g (R iR) before R, by conformal invariance. Since g (z) z 0 as K K + K − → z , we may choose R large enough that g (R iR) lies in the strip {z : R 1 Imz R 1}. → ∞ K + − < < + Therefore for R 1, > ImgK (z) ImgK (z) P(B hits R iR before R K ) R 1 < + ∪ < R 1 Substitute into (3.2.2) and take+ R to ﬁnd − → ∞ µ ¶ ImgK (Et ) P(E[t, ) K ) E . ∞ ∩ = ; = ImEt Taking t 0 and applying dominated convergence on the right-hand side gives P(E[0, ) K → ∞ ∩ = ) g 0 (0). ; = K

3.3 Reﬂected Brownian excursions

Fix θ (0,π) and consider a C-valued process B X iY started at a point z in the wedge W (θ) B ∈ = + {reiϕ : r 0 and 0 ϕ π θ}. We will define a new process Bˆ Xˆ iYˆ taking values in W (θ) by > < < − i(π θ) = + reflecting B horizontally off the ray {re − : r 0} and stopping it when it hits the real line. More > precisely, let c cotθ, let Yˆt Yt , and let = − = µ ¶ Xˆt Xt sup (cYt Xt ) 0 , = + s t − ∨ ≤ which is the unique process for which Xˆ cY and ` Xˆ X satisfies t ≥ t t = t − t (i) `t is nondecreasing and continuous, and (ii) The Stieltjes measure d` is supported on the set {t 0 : Xˆ cY }. t ≥ t = t See [10] for a proof of uniqueness. The relationship between Xt , cYt , and Xˆt is illustrated in Figure 3 below with toy functions Xt 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 (Bˆt )0 t τ be a reflected Brownian motion started at | − | ∈ ≤ ≤ R+ z W (θ). Then (Φ(Bˆt ))0 t τ is a time change of a reflected Brownian motion started at Φ(z). ∈ ≤ ≤ R+ i(π θ) Proof. Define the ray ρ {re − : r 0}. The hypotheses ensure that Φ0(z) 0 for all z ρ. (Note = > > ∈ 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 14

Xˆt

cYt

Figure 3: The reﬂected process (Xˆt )t 0. ≥ that the derivative exists because the boundary of W (θ) is straight in a neighbourhood of z ρ). R t ∈ Also, since d` is supported on {Bˆ ρ}, the process `˜ Φ0(B )d` is continuous, nondecreas- t t ∈ t = 0 s s ing, and has its Stieltjes measure d`˜ supported on {Φ(Bˆ) ρ}. Therefore, it sufﬁces to show that t ∈ Φ(Bˆ) `˜ is a time change of Brownian motion up to its exit time from H. Write z x i y and − = + Φ(x i y) u(x, y) i v(x, y). By Itô’s formula, we have + = + ∂u ∂u du(Xˆ ,Y ) (Xˆt ,Yt )d Xˆt (Xˆt ,Yt )dYt d`˜t = ∂x + ∂y − ∂u ∂u (Xˆt ,Yt )(d Xt d`t ) (Xˆt ,Yt )dYt d`˜t = ∂x + + ∂y − ∂u ∂u (Xˆt ,Yt )d Xt (Xˆt ,Yt )dYt , = ∂x + ∂y so the real part of Φ(Bˆ) `˜ is a local martingale. It follows from Lévy’s characterisation that Φ(Bˆ) `˜ − − is a time change of Brownian motion, as in the proof of Theorem 2.3.3.

π/(θ π) H W (θ) b (z) z − θ =

θ 0 0

Figure 4: A sample Brownian excursion reﬂected off the ray {re3πi/8 : r 0} in ≥ W (θ) and its image under bθ.

π/(π θ) Observe that that b (z) z − is a conformal map of W (θ) onto H which fixes the origin. θ = Definition 3.3.2. Let E be a Brownian excursion and let Eˆ 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(Eˆ). Theorem 3.3.3. Let E˜ be a reflected Brownian excursion with angle θ. Then for all K Q , we have ∈ + 1 θ/π P(E˜[0, ) K ) Φ0 (0) − , ∞ ∩ = ; = K where Φ (z) g (z) g (0) is the unique conformal isomorphism of H \ K H which satisfies K = K − K → Φ (0) 0 and Φ (z)/z 1 as z . K = K → → ∞ Proof. For r 0, let τ the hitting time of R ir . By Proposition 3.2.3, for all r 0, the Brownian ≥ r + > 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 15

˜ excursion started at Eτr and stopped at τR has the same law as a reflected Brownian motion started at E˜ , conditioned on the set {τ τ } and stopped at τ . Therefore, by Proposition 3.3.1, we may τr r < 0 R compute as in the proof of Proposition 3.2.6, · ¸ ImΦ˜ (E˜τ ) P(E˜[τ , ) K ) E r , (3.3.1) R ˜ ∞ ∩ = ; = ImEτr 1 for all r 0, where Φ˜ is the unique conformal map from W (θ)\b− (K ) W (θ) that fixes the origin > 1 → 1 and satisfies Φ˜ (z)/z 1 as z . Since b− Φ b satisfies these requirements, Φ˜ b− Φ b . → → ∞ θ ◦ K ◦ θ = θ ◦ K ◦ θ As z 0, we have → 1 Φ˜ (z) b− (Φ (b (z))) = θ K θ ³ ´1 θ/π π/(θ π) − Φ (z − ) = K ³ ´1 θ/π π/(θ π) 2π/(θ π) − Φ0 (0)z − O( z − ) , = K + | | 1 θ/π so Φ˜ 0(0) Φ0 (0) − . Therefore, taking the limit as r 0 in (3.3.1) and applying dominated con- = K → vergence, we find

1 π/θ P(E˜[0, ) K ) Φ0 (0) − . ∞ ∩ = ; = K 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 ∈ + = 5/8 P(E˜[0, ) K ) Φ0 (0) . (3.3.2) ∞ ∩ = ; = K

3.4 Loewner’s theorem

It turns out that there exists a simple random curve γ from 0 to in H which satisfies the same law ∞ (3.3.2) as a reflected Brownian excursion with angle 3π/8. Moreover, the Hausdorff dimension of this curve is known. We will use this observation to compute the almost-sure Hausdorff dimension of the Brownian frontier. The curve γ is an example of a class of curves called Schramm-Loewner evolutions (SLE). We will briefly sketch the framework in which SLE is defined. Proofs may be found in [5] or [8]. Let K be a compact H-hull and let gK be the unique conformal map of H \ K onto H for which gK (z) z 0 as z . Then there exists a unique aK R for which gK (z) 2 − → | | → ∞ ∈ = z a /z O( z − ) as z . The quantity a is denoted hcap(K ) and is called the half-plane + K + | | | | → ∞ K capacity of K . Define rad(K ) to be the radius of the smallest circle containing K whose centre is in

R. Let (Kt )t 0 be an increasing family of compact H-hulls, and define Kt,t h gKt (Kt h \ Kh). The ≥ + = + family (Kt )t 0 is said to satisfy the local growth property if Kt,t h 0 as h 0, uniformly as t ranges ≥ + → → T over 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 ξ . The function ξ : R+ R is called the Loewner t → transform of (Kt )t 0, and it is continuous. Loewner’stheorem describes a correspondence between ≥ ξ and the differential equation ∂ 2 gt (z) , g0(z) z, ∂t = g (z) ξ = t − t 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 T to be ≥ ∈ z the supremum of the times up to which a solution exists. For a proof of Loewner’s theorem, see [8]. Theorem 3.4.1. (Loewner) 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 16

0 0 0 κ 8/3 κ 4 κ 6 = = = Figure 5: Sample paths of numerical approximations of SLE for κ 8/3, 4, and κ = 6.

(i) Suppose that (Kt )t 0 is an increasing family of compact H-hulls satisfying the local growth ≥ property. Suppose further that (Kt )t 0 has been parametrised so that hcap(Kt ) 2t. Then ≥ = for each z H, the solution of the Loewner equation up to time T is given by t g (z). ∈ z 7→ Kt (ii) Let (ξt )t 0 be a continuous, real-valued process. For z H, deﬁne gt (z) to be the solution of ≥ ∈ Loewner’s equation up to time Tz , and deﬁne Kt {z : Tz t}. Then (Kt )t 0 is an increasing = ≤ ≥ family of compact H-hulls satisfying the local growth property, hcap(K ) 2t, the Loewner t = transform of (Kt )t 0 is ξ, and gK gt for all t 0. ≥ t = ≥ 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. ξ pκB , where B is a standard Brownian motion. It ≥ t = t t can be shown [5] that the complement Kt of Ht is the hull of a unique path γ. The random path γ is called SLEκ. Its behaviour exhibits phase transitions as κ increases, as stated in the following proposition (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 be the conformal map corresponding to γ . Deﬁne gˆ (z) cg 2 (z/c), ≥ t t t = t/c which is the family of conformal maps corresponding to cγt/c2 . Then ∂gˆt (z) 1 2/c 2 g˙t/c2 (z/c) , ∂t = c = g 2 (z/c) pκB 2 = gˆ (z) pκcB 2 t/c − t/c t − t/c and the second term in the denominator is a Brownian motion by Proposition 2.1.5(ii). 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 17

3.5 Restriction property of SLE8/3

Let us consider the image of a growing family of compact H-hulls under a conformal map. In particular, let (Kt )t 0 be a family of compact H-hulls satisfying the local growth property, let ξt T ≥ = h 0 Kt,t h be the driving function associated with Kt , and let (gt )t 0 be the corresponding family ≥ of conformal> + maps g : H\K H. Let Φ(z) a a z a z2 be a conformal map deﬁned in an t t → = 0 + 1 + 2 +··· H-neighbourhood U of ξ . Note that for Φ(U) H, we must have a ,a ,a ,... R and a 0. Let 0 ⊂ 0 1 2 ∈ 1 > T sup{t 0 : Kt U}. For 0 t T , deﬁne Kt∗ Φ(Kt ), at∗ hcap(Kt∗), and let gt∗ be the unique = ≥ ⊂ ≤ < = = 1 conformal map H \ K ∗ H with g ∗(z) z 0 as z . Finally, let Φ g − Φ g ∗. t → t − → → ∞ t = t ◦ ◦ t Proposition 3.5.1. With the preceding notation, we have

(i)( Kt∗)0 t T is a family of compact H-hulls satisfying the local growth property, ≤ < (ii)( ξ∗t )0 t T is the Loewner transform of (Kt∗)0 t T , ≤ < 2 ≤ < (iii) t a∗ has derivative a˙∗ 2Φ0 (ξ ) , and 7→ t t = t t (iv)( t,z) Φ (z) is differentiable in both coordinates with 7→ t Φ˙ (ξ ) 3Φ00(ξ ), and (3.5.1) t t = − t t Φ (ξ )2 4 ˙ 00t t Φ0t (ξt ) Φ000t (ξt ). (3.5.2) = 2Φ0t (ξt ) − 3

Sketch of proof. (i) and (ii) are easy to check. For (iii), observe that at∗ h at∗ hcap(gt∗(Kt∗ h \ 2 + − = 2 + Kt∗)) hcap(ξ∗t Φ0t (ξt ))(gt (Kt h Kt ) ξt ) Φ0t (ξt ) hcap(gt (Kt h Kt ) 2hΦ0t (ξt ) . To derive ≈ + + − − 1 = + − = (iv), assume first that t 0. Write f for g − . Differentiate f (g (z)) z with respect to t to find that = t t t t = 2ft0(z) f˙t (z) − . = z U − t a˙t∗ Also, note that g˙0∗(z) z ξ from Loewner’s equation, and apply the chain rule: = − 0∗ Φ˙ (z) g˙∗(Φ (f (z))) (g ∗)0(Φ (f (z)))Φ0 (f (z))f˙ (z) 0 = 0 0 0 + 0 0 0 0 0 0 2 2Φ0(ξ0) 2 (g0∗)0(Φ0(f0(z)))Φ0(f0(z))f00(z) = Φ0(z) Φ0(ξ0) − z U0 − 2 − 2Φ0 (ξ0) 2Φ0 (z) 0 0 . = Φ (z) Φ (ξ ) − z ξ 0 − 0 0 − 0 For 0 t T , we can apply the map gt to (Ks)t s T and gt∗ to (Ks∗)t s T to reduce to the case t 0. < < ≤ < ≤ < = Therefore 2 2Φ0t (ξt ) 2Φ0t (z) Φ˙ t (z) . (3.5.3) = Φ (z) Φ (ξ ) − z ξ t − t t − t Letting z ξ gives (3.5.1). Since the RHS of 3.5.3 is continuous, we can find Φ˙ 0(z) by differentiat- → t ing with respect to z to find 2 2Φ0t (ξt ) Φ0t (z) 2Φt (z) 2Φ00t (z) Φ˙ 0 (z) − , (3.5.4) t = (Φ (z) Φ (ξ ))2 + (z U )2 − z ξ t − t t − t − t which gives (3.5.2) by letting z ξ . → t α Proposition 3.5.2. The process Mt 1{t τA}Φ0t (ξt ) satisfies = < ·µ ¶ 2 µ ¶ ¸ dM Φ00(ξt ) (α 1)κ 1 Φ00(ξt ) κ 4 Φ000(ξt ) t t p − + t t κdBt 2 dt. (3.5.5) αMt = Φ0t (ξt ) + 2 Φ0t (ξt ) + 2 − 3 Φ0t (ξt ) 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 18

Proof. We apply Itô’s formula to ﬁnd dM Φ˙ (ξ ) Φ (ξ ) 1 · Φ (ξ )2 Φ (ξ )¸ t 0t t 00t t p 00t t 000t t dt κdBt (α 1) 2 κdt, αMt = Φ0t (ξt ) + Φ0t (ξt ) + 2 − Φ0t (ξt ) + Φ0t (ξt ) ˙ Equation (3.5.5) follows by substituting the expression in (3.5.2) for Φ0t (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 ∈ ± 5/8 P(γ[0, ) K ) Φ0 (0) , ∞ ∩ = ; = K where Φ (z) g (z) g (0) is the unique conformal isomorphism of H \ K H which satisﬁes K = K − K → Φ (0) 0 and Φ (z)/z 1 as z . K = K → → ∞ Proof sketch. Let D H \ K , and for ² 0, let D denote the set of all points in H whose distance = > ² from K is at least ². Deﬁne gt , gt∗, and Φt as in Proposition 3.5.1 with Φ ΦD² . Let τ be the hitting α = time of R K and Mt 1{t τ}Φ0t (ξt ) . Observe that choice κ 8/3 and α 5/8 makes (Mt )t 0 a ∪ = < = = ≥ local martingale, by Proposition 3.5.2. Since 0 Φ0t (ξt ) 1, (Mt )t 0 is a martingale. Note that by ≤ ≤ ≥ Proposition 3.2.6,

Φ0 (ξ ) P(E[0, ) g (D )), (3.5.6) t t = ∞ ⊂ t ² where E is a Brownian excursion started at ξt . Our goal is to show that Mt approaches the indicator of {τ } {γ[0, ) K } as t . Note that as t , g (H\D ) gets smaller and farther from = ∞ = ∞ ∩ = ; → ∞ → ∞ t ² the origin, which (when made rigorous) implies M 1 on the set {τ }. Also, on {τ }, γ t → = ∞ < ∞ τ lies on a ball B at least half of which is contained in H \ D . Therefore, there exists p 0 for which ² > a Brownian motion started at any point in B H has probability greater than p of exiting H\γ[0,t] ∩ on the boundary ( ,γ ) and also probability greater than p of exiting on (γ , ). It follows that −∞ τ τ ∞ g (B H) is contained in a proper cone with vertex ξ . By (3.5.6), this implies Φ0 (ξ ) 0. Thus τ ∩ τ τ τ = 5/8 Φ0 (0) M0 EMτ 1 P(γ D²) 0 P(γ H \ D² , ) P(γ D²). D² = = = · ⊂ + · ∩ ; = ⊂ Taking ² 0 gives the result. → Remark 3.5.4. The use of the term restriction for the property described in Theorem 3.5.3 comes from the fact that the theorem gives a quick proof of the restriction property of SLE8/3. See [8] for more details.

Remark 3.5.5. By Corollary 3.3.4 and Theorem 3.5.3, SLE8/3 has the same law as the right boundary of an RBE with angle 3π/8. We will use this observation to show that that they have the same Hausdorff dimension, from which it will follow that the dimension of the Brownian frontier is equal to 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 properties. For α 0, ² 0, and A C, deﬁne ≥ > ⊂ ½ ¾ α X∞ α [ H² (A) inf (diamDn) : A Dn; n,diamDn ² . = n 1 ⊂ n ∀ ≤ = α α As ² 0, the set of coverings {Dn}∞ decreases, so H² (A) increases. Therefore, lim² 0 H² (A) → n 1 & exists in [0, ]. = ∞ Proposition 3.6.1. H α(A) H β(A) 0 for all β α. < ∞ ⇒ = > 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 19

Proof. Calculate ½ ¾ β X∞ β [ H² (A) inf (diamDn) : A Dn; n,diamDn ² = n 1 ⊂ n ∀ ≤ ½ = ¾ X∞ β α α [ inf (diamDn) − (diamDn) : A Dn; n,diamDn ² = n 1 ⊂ n ∀ ≤ = ½ ¾ β α X∞ α [ ² − inf (diamDn) : A Dn; n,diamDn ² ≤ n 1 ⊂ n ∀ ≤ = β α α ² − H (A) 0 as ² 0. = ² → → Proposition 3.6.1 and establishes the existence of α such that H α(A) is equal to on the interval 0 ∞ (0,α ) and 0 on the interval (α , ). This critical value α is deﬁned to be the Hausdorff dimension 0 0 ∞ 0 of A, and is denoted dimH (A). α Lemma 3.6.2. H (A) 0 if and only if for all sequences ² 0+ and δ 0+, there exist sets U = m → m → m,j for which diamU ² for all j and P (diamU )α δ . m,j < m j m,j < m Proof. Since H α(A) increases as ² decreases, H α(A) 0 if and only if H α(A) 0 for all ² 0. The ² = ² = > equivalence of

H α(A) 0 for all ² 0 ² = > 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 all → 1 | − | ≤ | − | z,w C), then dim (f (A)) b− dim (A). ∈ H ≤ H

Proof. (From [5]) Let α dimH (A), let ²m 0+ and δm 0+, and let Um,j satisfy the conditions in > ˜ → → ˜ b P ˜ α/b Lemma 3.6.2. Then the sets Um,j f (V Um,j ) satisfy diamUm,j c²m and j (diamUm,j ) α/b P α α/b = ∩ ≤ ≤ c j (diamUm,j ) c δm. Thus dimH f (A) α/b for all α dimH A, which implies dimH (f (A)) 1 < ≤ > ≤ b− dimH (A).

[∞ Proposition 3.6.4. If A An, then dimH A supn dimH An. = n 1 = =

Proof. It is clear that dimH A supn dimH An. Conversely, let α supn dimH An. Let Un,m,j sat- m n ≥ P α m n > S isfy Un,m,j 2− − for all j and j diam(Un,m,j ) 2− − for all m,n. Let U˜m,j n Un,m,j . By < m < P α P m=n m the triangle inequality, diamU˜ 2− for all j. Also, (diamU ) 2− − 2− . Thus m,j < j m,j < n = dim A α, as desired. H ≤

Definition 3.6.5. We say that a process (Xt )t 0 satisfies the scaling property if (cXt/c2 )t 0 has the ≥ ≥ same distribution as (Xt )t 0, for all c 0. ≥ > Proposition 3.6.6. Suppose that (Xt )t 0 satisfies the scaling property. For 0 s t , the ran- ≥ < < < ∞ dom variable dim X [s,t] has the same distribution as dim X [0, ). H H ∞ Proof. Write X (0, ) S X [qs,qt], where the union ranges over all rationals q 0. By the scaling ∞ = q > hypothesis, the distribution of the dimension of X [qs,qt] does not depend on q. By Proposition 3.6.4, the distribution of dim X [qs,qt] is equal to that of X (0, ). H ∞ 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 20

Proposition 3.6.7. Suppose that (Xt )t 0 satisﬁes the scaling property, and let d 0. Then ≥ ≥ dim X [0, ) d almost surely H ∞ = implies that dim X [σ,τ] d almost surely for all random times σ and τ with 0 σ τ . H = < < < ∞

Proof. Suppose ﬁrst σ and τ take on only countably many values, say {sn}n∞ 1 and {tn}n∞ 1. Then = = ¡ ¢ X∞ ¡ ¢ P dimH (γ[σ,τ] d) P dimH X [sm,tn] d and (σ,τ) (sm,sn)) = = m,n 1 = = = X∞ ¡ ¢ P (σ,τ) (sm,tn) 1. = m,n 1 = = = 1 1 S For general σ and τ, deﬁne σn B n− nσ and τn B n− τn . Since X [σ,τ] n X [σn,τn], the d e b c = result follows from Proposition 3.6.6.

Remark 3.6.8. To make statements about distributions of Hausdorff dimensions of random sets rigorous, we need a σ-algebra on a collection of subsets of C for which the map which sends a set to its Hausdorff dimension is measurable. It is possible to do this, for example, on the set of compact subsets of C by taking the Borel sets corresponding to the Hausdorff metric µ ¶ µ ¶ d(A,B) sup inf a b sup inf a b . = a A b B | − | ∨ b B a A | − | ∈ ∈ ∈ ∈ In Section 3.8, we will deﬁne an ad hoc σ-algebra which respects taking the dimension of the boundary rather than the dimension of the set itself. Since we will apply Propositions 3.6.6 and 3.6.7 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 given in [3]. Observe that for κ 8, the dimension of SLE is trivially 2, since the path is space filling. ≥ κ Also, SLE is given by the simple formula γ(t) 2i pt, so the dimension of SLE is 1. Between κ 0 0 = 0 = and κ 8, it turns out that that the relationship between κ and the Hausdorff dimension of SLE = κ is linear. Theorem 3.7.1. The Hausdorff dimension of SLE is almost surely 2 ¡1 κ ¢. κ ∧ + 8 Heuristic proof of upper bound. We begin with a general observation about random sets. Recall that f (²) Θ(g(²)) as ² 0 if f is bounded above and below by a constant multiple of g as ² = 2 → 2 →d 0. It takes Θ(²− ) balls of radius ² to cover a given (bounded) region in R , and of these Θ(²− ) are needed to cover K . Therefore, we expect that the probability that an ²-neighbourhood of z 2 d intersects K should take the form ² − f (z). As previously observed, the theorem is trivially true when κ 8. So fix 0 κ 8, and let P(ζ,ζ,²,a) ≥ ≤ < denote the probability that an SLE path γ started at a visits the ²-neighbourhood of ζ H. We will κ ∈ derive an expression for P by showing that it solves a certain partial differential equation. To this end, imagine evolving the Loewner flow for an infinitesimal time dt. Under the conformal map 2dt gdt , a maps to a0 a pκdBt , ζ maps to ζ0 ζ ζ a (from the Loewner differential equation), = + = 2 + − and ² maps to ²0 ² gdt0 (ζ) ² 2²Re((ζ a)− )dt. To obtain the last expression, differentiate 2dt = | | = − − ζ ζ ζ a and observe that for small z, 1 z 1 Rez. Recall that that gdt (γ) is an SLEκ started 7→ + − | − | ≈ − at a0. Therefore, an SLEκ started from a0 visits visit the ²0-neighbourhood of ζ0 if and only if γ visits 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 21 the ²-neighbourhood of ζ. So P satisfies " Ã !# 2dt 2dt ¡ 2¢ P(ζ,ζ,²,a) E P ζ ,ζ ,² 2²Re (ζ a)− dt,a pκdBt = + ζ a + ζ a − − + − − Taylor expand the right-hand side, to first order in the first three arguments and to second order in the fourth argument. Simplify using EdB 0 and E(dB )2 dt, and subtract P(ζ,ζ,²,a) from t = t = both sides to obtain Ã 2 ! 2 ∂ 2 ∂ κ ∂ ¡ 2¢ ∂ 2 Re (ζ a)− ² P 0. ζ a ∂ζ + ζ a ∂ζ + 2 ∂a2 − − ∂² = − − To simplify this differential equation, let a 0 without loss of generality. Also, switch to Cartesian = coordinates ζ x i y and use the identities = + ∂ ∂ ∂ , and ∂x = ∂ζ + ∂ζ ∂ ∂ ∂ i i ∂y = ∂ζ − ∂ζ to rewrite 2 ∂ 2 ∂ 2x ∂ 2y ∂ . ζ ∂ζ + = x2 y2 ∂x − x2 y2 ∂y ζ ∂ζ + + Finally, observe that P(x, y,²,a) P(x a, y,²,0), which implies ∂P ∂P . Altogether, we find = − ∂a = − ∂x µ 2x ∂ 2y ∂ κ ∂2 2(x2 y2) ∂ ¶ − ² P 0 (3.7.1) x2 y2 ∂x − x2 y2 ∂y + 2 ∂x2 − (x2 y2)2 ∂² = + + + We expect that P scales as some power of ²/ 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 µ ¶2 d 2 α ² − ³ y ´ 2 d d 2 2β 2 2 β 2 2 as ² − y − − (x y ) and substitute in (3.7.1) to obtain px2 y2 x y + + + 2 d 2β 2 2 2 β 2 d ¡ 2 2¢ y− + − (x y )− + ² − (κ 8 2κβ)βx (4d 8 κβ)y 0. − + − − + − − = Solving the system κ 8 2κβ 0, − − = 4d 8 κβ 0 − − = gives d 1 κ , as desired. = + 8 It is worth emphasizing that the ideas in this derivation are only sufficient to give an upper bound 2 d on the dimension, since a priori the form ² − f (z) could result from some SLE sample paths intersecting many balls, whereas ‘most’ paths intersect fewer [1]. Estimates on the probability of the path intersecting two arbitrary disks can be used to prove dim (SLE ) 2 (1 κ/8). H κ ≥ ∧ +

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] and ⊂ −∞ its complement H \ F is simply connected. Let Ω denote the set of all such F , equipped with the + σ-algebra Σ generated by E {SK : K Q }, where SK B {F : F K }. Observe that the law of + = ∈ + ∩ = ; F is determined by the values of P(F K ) for K Q , since SK SK SK K shows that E is a ∩ = ; ∈ + 1 ∩ 2 = 1∪ 2 π-system. Denote by ∂ F the right boundary ∂(H \ F ) \ (0, ) of the ﬁlling F . r ∞ 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 22

Proposition 3.8.1. The map F dimH (∂r F ) from (Ω ,Σ ) to ([0,2],B([0,2]) is measurable. 7→ + + Proof. Define U to be the set of all balls in C which intersect H, have a rational radius, and whose centres has rational coordinates. Enumerate the elements U ,U ,... of U. Let B U, and define 1 2 ∈ E(B) {F : F B , }. We will show that E(B) Σ . First, observe that if V is a bounded simply = ∩ ; ∈ + connected open set for which R ∂(H V ) is a nonempty subset of (0, ), then E(V ) Σ . To ∩ ∩ ∞ ∈ + see this, approximate V with a countable sequence of compact subsets of V . Define P to be the collection of all finite unions of elements of U. To see that P is countable, observe that [∞ P {A : A is a union of sets in {U1,U2,...,Un}}. = n 1 = Enumerate the elements P ,P ,... of P. If B intersects ∂ F , then there exists P P for which P \ B 1 2 r ∈ is connected and ∂r F does not intersect the closure of any of the other balls whose union is P [9]. Therefore,

[∞ E(B) E(Pn)\ E(Pn \ B), = n 1 = 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, ⊂ ½ ¾ α X∞ α [∞ H˜ (A) lim inf diam(Dn) : A Dn, diamDn 1/k, Dn U for all n , = k n 1 ⊂ n 1 ≤ ∈ →∞ = = α α α α α α we have 2− H˜ (A) H (A) H˜ (A). In particular, H˜ is nonzero if and only if H is nonzero, so ≤ ≤ dim (A) inf{α : H˜ α(A) 0}. Writing H = = ˜ α X∞ α H (∂r F ) lim diam(Un) 1E(Un )1diamUn 1/k = k n 1 ≤ →∞ = shows that (α,V ) H˜ α(V ) is measurable, and finally 7→ ½ ¾ α dimH (∂r F ) inf α1{H (∂r F ) 0} = α Q = ∈ + shows that F dim (∂ F ) is measurable. 7→ H r Proposition 3.8.2. The right boundary of a reflected Brownian excursion with angle 3π/8 has Hausdorff 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 excursion with angle 3π/8. By Theorem 3.7.1, µ (S) 1. By Corollary 3.3.4 and Theorem 3.5.3, we have 1 = µ1 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 sets of the form

©{A : A F }: F Cˆ is closed and Cˆ \ F is simply connectedª. ∩ = ; ⊂ Here Cˆ denotes the extended complex plane C { }. Deﬁne the frontier fr A of a set A Cˆ to be ∪ ∞ ⊂ the boundary of the complement of the unbounded component of C \ A. 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 23

a(A,x)

∂l,x

A 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 probabilities P(A F ), as shown by the next proposition. ∩ = ; Proposition 3.8.4. The map A dim (fr A) is A-measurable. 7→ H Proof. Follow the proof of Proposition 3.8.1, with the modification that U is defined to be the set of all balls in Cˆ with rational radius and centre either at or at a point with rational coordinates. © ∞ 1ª (Recall that a ball centred at is a set of the form z : z r − 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 reiθ : r 0}, = + ≥ and define the translation σ (z) z ξ. Define a(A,x) x i sup{Imz : Rez x and z A} and ξ = − = + = ∈ b(A,x) x i inf{Imz : Rez x and z A}. Choose y large enough that σ i y (A) H, let ξ x i y, = + = ∈ − ⊂ = + and define

³ 1 ³ ³ ´ ³ ´´´ ∂ A A σ− ∂ σ A ρ a(A,x),π/2 ρ b(A,x), π/2 , r,x = ∩ ξ ◦ r ◦ ξ ∪ ∪ − and deﬁne ∂l,x analogously (see Figure6).

Proposition 3.8.6. The frontier of a compact set A can be written as fr A ¡∂ A¢ ¡∂ A¢. = l,x ∪ r,x 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. By ∞ continuity, 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 ¡∂ A¢ ¡∂ A¢ fr A. ∞ l,x ∪ r,x ⊂ Conversely, if z fr A, then let γ be a path connecting z to in C \ A. If γ does not intersect ³ ´ ∈ ³ ´ ∞ 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 intersects ∪ − ρ (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 t . By compactness, there exists → 0 ² 0 for which the strip > {z : x Rez γ(t ²) and Imz Ima(A,x)} < < 0 − > lies in C \ A. Deﬁne a new path γ∗ which is equal to γ up to time t ²/2, follows a vertical ray out 0 − 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 24 to a neighbourhood of in C\ A on the interval [t ²/2,t ²/2], and follows a circular arc down ∞ 0 − 0 + to R+ on the interval [t ²/2,1]. 0 + d Lemma 3.8.7. Suppose that (Kt )t 0 is a family of random compact subsets of C for which Kt ≥ = ptK for all t 0. Let τ be a random time whose range is countable. Then dim K d almost 1 > H τ = surely if and only if dim K d almost surely. H 1 = Proof. Suppose that dim K d almost surely. By the scaling hypothesis, the distribution of H 1 = dim K does not depend on t, so dim K d almost surely for all t 0. Enumerate the elements H t H t = > in the range of τ as {tn}n∞ 1. Then = µ ¶ [∞ ¡ ¢ P(dimH Kτ d) P {dimH Ktn d} {τ tn} = = n 1 = ∩ = = X∞ ¡ ¢ P {dimH Ktn d} {τ tn} = n 1 = ∩ = = X∞ P(τ tn) 1. = n 1 = = = Conversely, suppose that dim K d almost surely. If Ω is written as a disjoint union S Ω and H τ = n n A is an event whose probability is less than 1, then there exists a natural number n for which P(A Ωn) P(Ωn). So if P(dimH K1 , d) 0, we have ∩ < > X∞ ¡ ¢ P(dimH Kτ d) P {dimH Ktn d} {τ tn} = = n 1 = ∩ = = X∞ P(τ tn) 1, < n 1 = = = a contradiction.

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

Definition 3.8.8. Let (Xt )t 0 be a continuous random process in C, let σ be a random time, and let ≥ r be a positive random variable. The neighbourhood rational random time associated with σ and r 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 be the smallest. Theorem 3.8.9. If B is a Brownian motion in C, then dim frB[0,1] 4/3 almost surely. H = ˆ 1 5/8 H Proof. Let E be the image of a reflected Brownian excursion under the map bθ− (z) z from to 1 = the wedge W (3π/8). It is easy to verify that the right boundary of Eˆ[0, ) is b− (∂ (bθ(Eˆ[0, )))). ∞ θ r ∞ ˆ 1 Let γ˜(t) be a parametrisation of ∂r E[0, ), which has the same law as bθ− γ where γ is an SLE8/3 1∞ ◦ by Theorems 3.3.3 and 3.5.3. Also, bθ− preserves Hausdorff dimension by Propositions 3.6.3 and 3.6.4. Therefore, dim ∂ Eˆ[0, ) 4/3 almost surely. H r ∞ = Define the stopping times σ inf{t 0 : ReEˆ 2}, τ inf{t σ : ReEˆ 1}. By Proposition 2.1.8, = ≥ t = = ≥ t = 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 25

r τ σ1

σ2

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

σ and τ are ﬁnite almost surely. Let σ1 and σ2 be random times deﬁned by σ inf{σ t τ : Eˆ sup{ImEˆ : ReEˆ 2 and σ s τ}} 1 = ≤ ≤ t = s s = ≤ ≤ σ inf{σ t τ : Eˆ inf{ImEˆ : ReEˆ 2 and σ s τ}} 2 = ≤ ≤ t = s s = ≤ ≤ Let r inf{ Eˆ Eˆ : t is between σ and σ }, and let τ0 be the neighbourhood rational random = | τ − t | 1 2 time associated with τ and r . (Note that r 0 by compactness). By the continuity of Eˆ, > ∂ Eˆ[σ,τ] ∂ Eˆ[σ,τ0]. (3.8.1) r,1 = r,1

(1,n)

1 ∆(Eˆτ,n− ) θ

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

Proposition 2.1.9 shows that dim ∂ Eˆ[σ,τ] d intersects (g γ)(I) on an interval I of positive H r,2 ≤ ◦ length, which implies dim ∂ Eˆ[σ,τ] 4/3 almost surely by Proposition 3.6.7. Suppose that the H r,2 ≥ event C B {dimH ∂r,2Eˆ[σ,τ] 4/3} occurs with positive probability. Let ∆(z,r ) denote the open > disk of radius r centred at z. For z with real part 1 and n N, deﬁne the domain U (z,θ) (see ∈ n Figure8) by

U (z,θ) ∆(Eˆ ,1/n) (W (θ)\ ([1, ) [0,n])). n = τ ∪ ∞ × Define the events D {Eˆ[τ, ) U (Eˆ ,θ))}. By Propositions 2.5.3 and 2.2.6, P(D ) 0 for all n = ∞ ⊂ n τ n > n N. By the monotone convergence theorem, P(C {Eˆ(σ,τ) U (Eˆ ,θ) }) 0 for large enough ∈ ∩ ∩ n σ = ; > 3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER 26 n. By the strong Markov property, C and D are independent. Therefore, P(C D ) 0. Moreover, n ∩ n > we may cover ∂r,2Eˆ(σ,τ] with countably many open disks which do not intersect Eˆσ. If ∂r,2Eˆ(σ,τ] has dimension greater than 4/3, then its intersection with at least one of these disks has dimension greater than 4/3 as well, by Proposition 3.6.4. Index the disks by N and define ∆ to be the disk with least index whose intersection with ∂r,2Eˆ[σ,τ] has dimension greater than 4/3. With positive probability, Eˆ[0,σ] does not intersect ∆. Since Eˆ[0,σ] is independent of (Eˆσ t Eˆσ)t 0, we find 1 + − ≥ that ∆ ∂ Eˆ[σ,τ] coincides with (b− γ)[0, ) on an interval of positive length, with positive ∩ r,2 θ ◦ ∞ probability. Again by Proposition 3.6.7, this is a contradiction. Therefore, dim ∂ Eˆ[σ,τ] 4/3 H r,1 = almost surely. From (3.8.1), we also have dim ∂ Eˆ[σ,τ0] 4/3 almost surely. H r,1 = Since Eˆ[σ,τ0] has the same law as a Brownian excursion started at Eˆσ, up to time τ0, we conclude by Lemma 3.8.7 (along with a suitable translation) that dim ∂ E[0,T ] 4/3 almost surely for a H r,0 = Brownian excursion E started at i and a stopping time T whose range is countable. Let T denote the neighbourhood rational random stopping time associated with the hitting time of R 3i/2 and + radius r 1/2, and let T 0 be the neighbourhood rational random time associated with the hitting = time of R i/2 and radius r 1/2. By Proposition 3.2.3, E[0,T T 0] has the same distribution as + = ∧ B˜[0,T T 0], where B˜ is a Brownian motion started at i and conditioned to hit R 2i before R. Thus ∧ + dimH ∂r,0B˜[0,T T 0] 4/3 almost surely. Since B˜ is a conditioned Brownian motion, we have ¡ ∧ = ¢ ¡ ¢ 1 P dim ∂ B˜[0,T T 0] 4/3 2P dim ∂ B[0,T T 0] 4/3 and B hits R 2i before R , = H r,0 ∧ = = H r,0 ∧ = + which by symmetry implies P(dim ∂ B[0,T T 0] 4/3) 1. By Lemma 3.8.7, dim ∂ B[0,1] H r,0 ∧ = = H r,0 = 4/3 almost surely. By Proposition 3.8.6 and symmetry, dim frB[0,1] 4/3 almost surely. H = REFERENCES 27

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