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

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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 analy- sis 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 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 defined ¸ on a probability space (­,F,P) which satisfies the following conditions. (i) B 0 almost surely, 0 Æ (ii) for almost all ! ­, t B (!) is continuous, and 2 7! t (iii) for all s,t 0, Bt s Bt is independent of (Bu)0 u t and has distribution N(0,h), ¸ Å ¡ · · 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 (­,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 Æ (ii) for almost all ! ­, t B (!) is continuous, 2 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). ¸ Å ¡ 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 ZÅ,d D } be a countable collec- n Æ 2 2 2 n,t 2 2 n tion of iid standard normal random variables. Set B (0) 0, B (0) Z , and B (0) t Z (i.e., linearly 0 Æ 1 Æ 0,1 t Æ 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 Æ t Dn 1. For t Dn \ Dn 1, set t ¡ t 2¡ , t Å t 2¡ , and 2 ¡ 2 ¡ Æ ¡ Æ Å (n 1) (n 1) B ¡ B ¡ (n 1) t ¡ t Zn,t B Å Å Å t Æ 2 Å 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 copies2 of Brownian 1 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 · · Æ ¸ 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 Æ ^ ¸ Definition 2.1.4. The Wiener space W is defined to be the space C([0, ),R) of continuous real- 1 valued functions on [0, ), with the metric 1 X1 n d(f ,g) 2¡ sup f (x) g(x) . Æ n 1 0 x n j ¡ j Æ · · We equip W with its Borel σ-algebra W B(W ). (We use the notation B(X ) for the Borel σ-algebra Æ on a topological space X .) We may think of a Brownian motion B as a map ­ W which sends ! to the continuous function ! t B (!). 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). È t/c (iii) The process starting at 0 and equal to tB for t 0 is a Brownian motion (time inversion). 1/t È 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 Æ s t Æ ^ £ ¤ 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 Æ s/c t/c Æ ^ Æ ^ for (ii), and E[(sB )(ctB )] st E[B B ] st(1/s 1/t) min(s,t) 1/s 1/t Æ 1/s 1/t Æ ^ Æ for (iii). The only thing left to prove is continuity at zero for (iii). This follows since (tB1/t )t 0 È and (Bt )t 0 are both mean-zero Gaussian processes on C((0, ),R) and therefore have the same È 1 distribution as C((0, ),R)-valued random variables. Thus 1 limsup Bt 0, a.s. limsup tB1/t 0, a.s. t 0 j j Æ ) t 0 j j Æ ! ! Definition 2.1.6. Let (­,F,P,(Ft )t 0) be a filtered probability space. An RÅ-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 · 2 ¸ ¸ T , the σ-algebra FT is defined by F {A F : A {T t} F for all t 0}. T Æ 2 \ · 2 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 ¸ ¸ Å ¡ BT )t 0 is a Brownian motion with respect to the filtration (FT t )t 0 and is independent of FT .
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