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 Analysis
2.1 One-dimensional Brownian motion . . . . . . . . .
2.2 Brownian motion in Rd . . . . . . . . . . . . . . . .
2.3 Conformal invariance of planar Brownian motion
2.4 Applications to complex analysis . . . . . . . . . . .
2.5 Applications to planar Brownian motion . . . . . .
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3 Schramm-Loewner evolutions and the dimension of the Brownian frontier
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Brownian excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Reflected Brownian excursions . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Loewner’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Restriction property of SLE8/3 . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Hausdorff dimension of SLE . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Hausdorff dimension of the Brownian frontier . . . . . . . . . . . . . . . .
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1 INTRODUCTION
This essay explores the interplay between complex analysis and planar Brownian motion. In Section 2, 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. Section 3
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, SchrammLoewner 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’s book [11] and Nathanaël Berestycki’s notes on Stochastic 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 Brownian excursions comes from Lawler, Schramm, and Werner’s paper on conformal restriction measures [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 (B t )t ≥0 is a real-valued stochastic process defined
on a probability space (Ω, F, P) which satisfies the following conditions.
(i) B 0 = 0 almost surely,
(ii) for almost all ω ∈ Ω, t 7→ B t (ω) is continuous, and
(iii) for all s, t ≥ 0, B t +s − B t is independent of (B u )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 (B t )t ≥0 is
an (Ft )t ≥0 -Brownian motion if the following conditions hold.
(i) B 0 = 0 almost surely,
(ii) for almost all ω ∈ Ω, t 7→ B t (ω) is continuous,
(iii) (B t )t ≥0 is adapted to (Ft )t ≥0 (i.e., B t is Ft -measurable for all t ≥ 0), and
(iv) for all s, t ≥ 0, B t +s − B t is independent of Ft and has distribution N(0, h).
If (B t − X )t ≥0 is a standard Brownian motion for some random variable X , we say that (B t )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 (B t )t ≥0 satisfying conditions (i)-(iii) in Definition
2.1.1.
©
ª
Idea of proof. Define D n = 0, 21n , 22n , 23n , . . . , 1 , and let {Zn,t : n ∈ Z+ , d ∈ D n } be a countable collection of iid standard normal random variables. Set B 0(0) = 0, B 1(0) = Z0,1 , and B t(0) = t Z0,1 (i.e., linearly
2
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
interpolated between the values at 0 and 1). Define B t(n−1) inductively so that B t(n) = B t(n−1) for
t ∈ D n−1 . For t ∈ D n \ D n−1 , set t − = t − 2−n , t + = t + 2−n , and
+ B t(n−1)
B t(n−1)
−
+
Zn,t
2
2n
In other words, we have added a Gaussian fluctuation with the appropriate variance at the new
dyadic times. Again, extend B t(n) to [0, 1] by linear interpolation. Using some elementary estimates,
one can show that B t(n) is uniformly Cauchy, from which it follows that there
´ a uniform limit
³ exists
B t(n+1) =
+
B t . This limit inherits independent, normally distributed intervals from B t(n)
t ∈D n
. We obtain a
Brownian motion on [0, ∞) by summing a countable collection of independent copies of Brownian
motion 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
(t k )1≤k≤n is called a Gaussian process. If E X t = 0, then (X t )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 E X t X s = 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 realvalued functions on [0, ∞), with the metric
∞
X
2−n sup | f (x) − g (x)|.
d(f ,g) =
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 7→ B t (ω). The image measure of B is called the Wiener measure and is denoted W.
Proposition 2.1.5. Let (B t )t ≥0 be a Brownian motion.
(i) (−B t )t ≥0 is a Brownian motion.
(ii) If c > 0, then cB t /c 2 is a Brownian motion (scaling).
(iii) The process starting at 0 and equal to t B 1/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 jointly
Gaussian finite-dimensional distributions from (B t )t ≥0 . To check the covariance, we compute
E(−B s , −B t ) = E(B s , B t ) = s ∧ t for (i),
£
¤
£
¤
E (cB s/c 2 )(cB t /c 2 ) = c 2 E B s/c 2 B t /c 2 = c 2 (s/c 2 ∧ t /c 2 ) = s ∧ t
for (ii), and
E [(sB 1/s )(c t B 1/t )] = st E [B 1/s B 1/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 (t B 1/t )t >0
and (B t )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
lim sup |B t | = 0, a.s.
t →0
⇒
lim sup |t B 1/t | = 0, a.s.
t →0
Definition 2.1.6. Let (Ω, F, P, (Ft )t ≥0 ) be a filtered probability space. An R+ -valued random variable 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
FT = {A ∈ F : A ∩ {T ≤ t } ∈ Ft for all t ≥ 0}.
3
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
The following theorem says that a Brownian motion ‘restarted’ at a stopping time T is a new Brownian motion independent of the past. For a proof, see [2].
Theorem 2.1.7. (Strong Markov Property of Brownian motion) Let (B t )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 (B T +t −
B T )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, lim supt B t = ∞ and lim supt B t = −∞.
Proof. (From [11]). Let X = supt ≥0 B t . 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(X 1 ≤ 0 and X 1+t ≤ 0 ∀ t ≥ 0)
= P(X 1 )P(X 1+t ≤ 0 ∀ t ≥ 0)
which implies p = 0.
=
(by the Markov property)
1
p,
2
Proposition 2.1.9. For all ² > 0, the set {B t : 0 ≤ t ≤ ²} almost surely contains both positive and
negative numbers.
Proof. Apply Proposition 2.1.5(iii) to Proposition 2.1.8. Since B t almost surely takes on arbitrarily
large and small values as t ranges over the interval (²−1 , ∞), we find that (t B 1/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 (M t )t ≥0 is defined to be an adapted process for which E|M t | < ∞ for all t ≥ 0 and
E(M t − M s |Fs ) = 0,
∀ 0 ≤ s ≤ t < ∞.
Also, recall that a collection X of random variables is said to be uniformly integrable if
sup E(|X |1|X |>K ) → 0 as K → ∞.
X ∈X
Theorem 2.1.10. (Optional Stopping Theorem) Let (M t )t ≥0 be a continuous adapted process for
which E|M t | < ∞ for all t ≥ 0. Then the following are equivalent.
(i) (M t )t ≥0 is a martingale.
(ii) (M T ∧t )t ≥0 is a martingale for all bounded stopping times T .
(iii) E(X T |FS ) = X S∧T for all bounded stopping times S, T .
(iv) E X 0 = E X T for all bounded stopping times T .
Also, if (M t )t ≥0 is a uniformly integrable martingale, then E(X T |FS ) = X S∧T for all stopping time
S, T .
The following proposition allows us to give bounds for the probability that sup{|B t | : 0 ≤ t ≤ ²} < x.
The proof is from [2].
Proposition 2.1.11. Define S t = sup0≤s≤t B s . Then for all t ≥ 0, S t has the same distribution as |B t |.
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
4
Proof. For all t ≥ 0 and a ≥ 0, we have
P(S t ≥ a) = P(S t ≥ a and B t ≤ a) + P(S t ≥ a and B t > a)
= P(B t ≥ a) + P(B t > a)
= 2P(B t ≥ a) = P(|B t | ≥ a).
The step P(S t ≥ a and B t ≤ a) = P(B t ≥ a) comes from the reflection principle [2], which asserts
that for all a, b ∈ R, P(S t ≥ a and B t ≤ b) = P(B t ≥ 2a − b).
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 (M t )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 (M Tn ∧t )t ≥0 is a martingale for all n. For every continuous local martingale, there exists a
unique continuous adapted nondecreasing process, denoted [M ]t and called the quadratic variation of (M )t ≥0 , for which (M 2 − [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 (B t )t ≥0 is
t.
Proposition 2.1.12. Let 0 < a < b, let B be a Brownian motion started at a, and let τr = inf{t ≥ 0 :
B t = 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
(B t )t ∧τ0 ∧τb to find that a = EM 0 = EM τ0 ∧τb = pb + (1 − p)0. For the second equality, note that
B t2∧τ0 ∧τb − t ∧τ0 ∧τb is a martingale. Therefore, EB t2∧τ0 ∧τb − Et ∧τ0 ∧τb = EB 02 = a 2 . Take t → ∞ and
apply dominated convergence for the first term and monotone convergence for the second term
to find
b 2 (a/b) + 0 − Eτ0 ∧ τb = a 2 ,
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 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 rotationally 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
i
j
[M , M ]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 U B is a Brownian motion.
Proof. Let u 1 , u 2 , . . . , u d be the rows of the matrix U . As a linear combination of continuous local
martingales, the components of U B are continuous local martingales. Also, the i th component
of U B is (U B )i = u iT B , so [(U B )i , (U B ) j ]t = u iT 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.
5
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
Given a d -dimensional Brownian motion (B t )t ≥0 , it is often useful to show that ( f (B t ))t ≥0 is a martingale 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
¯∂f
¯ X ¯ ∂f
¯ X
¯
¯
¯
| f (t , x)| + ¯¯ (t , x)¯¯ +
(t , x)¯¯ +
(t , x)¯¯ ≤ K exp(K (t + |x|),
(2.2.1)
¯
¯
∂t
i =1 ∂x i
i , j =1 ∂x i ∂x j
for all (t , x) ∈ [0, ∞) × Rd . Then
C t B f (t , B t ) − f (0, B 0 ) −
Z tµ
0
¶
∂ 1
+ 4 f (s, B s ) d s
∂t 2
Pd
is a martingale. Here 4 denotes the Laplacian i =1 ∂2 /∂x i2 . In particular, if f is harmonic (that is,
4 f is identically zero), then f (t , B t ) is a martingale.
Proof. First we will show that E(C t −C s ) = 0 for 0 < s ≤ t . Denote by p t (x) = exp(−|x|2 /2t )/(2πt )−d /2
the density of B t . We compute
µ
¶
¶
Z tµ
∂ 1
E(C t −C s ) = E f (t , B t ) − f (s, B s ) −
+ 4 f (u, B u ) d u
∂t 2
s
µ
¶
Z
Z tZ
¡
¢
∂ 1
=
p t (x) f (t , x) − p s (x) f (s, x) d x −
p u (x)
+ 4 f (u, x) d x d u
∂t 2
Rd
s Rd
Z
Z tZ
¡
¢
∂
=
p t (x) f (t , x) − p s (x) f (s, x) d x −
(p u (x) f (u, x)) d x d u,
Rd
s Rd ∂u
R
R
where we have used integration by parts twice to rewrite p u (x)4 f (u, x) d x as 4p u (x) f (u, x) d x.
Here we require the growth condition (2.2.1) so that the product p u (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
4p u (x) = ∂p u (x)/∂u. Now we may interchange the integrals in the second term, since the mod2
ulus of the integrand is bounded. An appeal to the fundamental theorem of calculus completes
the proof that E(C t −C s ) = 0.
Now we will to take s → 0 to show that E(C t ) = 0. First, observe that C t −C s → C t p
pointwise as s → 0.
By Propositions 2.1.11 and 2.1.5(ii), we have E(sup0≤u≤t |B u | ≥ a) ≤ 2E(|B 1 | ≥ a/ t ). Thus
¯
¯
¶
Z tµ
¯
¯
1
∂
¯
|C t −C s | = ¯ f (t , B t ) − f (s, B s ) −
+ 4 f (u, B u ) d u ¯¯
∂u 2
s
¯
¶
Z t ¯µ
¯ ∂
¯
1
¯
¯ du
≤ | f (t , B t )| + | f (s, B s )| +
+
4
f
(u,
B
)
u
¯ ∂u 2
¯
s
µ µ
¶¶
≤ (2 + t )K exp K t + sup |B u | ,
0≤u≤t
which is integrable because
µ
¶
µ
µ
¶
¶
∞
X
E exp K sup |B u | ≤ 1 +
P exp K sup |B u | ≥ n
0≤u≤t
n=1
∞
X
0≤u≤t
µ
¶
log n
≤ 1+2
P |X 1 | ≥ p
K t
n=1
p
∞
X K t
exp(−(log n)2 /2K 2 t ) < ∞.
≤ 1+4
p
2π log n
n=1
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
6
−1
We have used the estimate E(X 1 ≥ x) ≤ x2π exp(−x 2 /2) for the tail of the normal distribution. Thus
E(C t ) = 0 for all t . By the Markov property of Brownian motion, this gives E(C t − C s |Fs ) = 0, as
desired.
Theorem 2.2.5. Planar Brownian motion is neighbourhood-recurrent but does not visit a prespecified point. More precisely, let B t be a planar Brownian motion started at z 0 ∈ C. Then for
all open U ⊂ C, the set
{t ≥ 0 : B t ∈ U }
(2.2.2)
is almost surely unbounded, and for all w , z 0 ,
{t ≥ 0 : B t = w} = ;,
almost surely.
(2.2.3)
Proof. Without loss of generality, suppose z 0 , 0. We will show that with probability 1, the Brownian 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| ≤ b}
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 (B t ) is a martingale.
∂2 f
∂2 f
y 2 −x 2
x 2 −y 2
Also, 4 f = ∂x 2 + ∂y 2 = y 2 +x 2 + y 2 +x 2 = 0 on {z : a < |z| < b}. Applying the optional stopping theorem
to the uniformly integrable martingale f (B t ) with the stopping time τ = inf{t ≥ 0 : |B t | ∈ {a, b}}, we
find that the probability p that the Brownian motion hits {z : |z| = a} before {z : |z| = b} solves
Thus p =
log b−log z 0
.
log b−log a
log |z 0 | = p log a + (1 − p) log b.
Letting b → ∞ gives (2.2.2). Also, letting a → 0 gives (2.2.3), since
P(B hits 0) =
=
∞
[
n=1
∞
[
P(B hits 0 before {z : |z| = n})
\
n=1 m>1/|z 0 |
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 at
z ∈ Rd never visits the closed ball of radius 0 < r < |z| centered at the origin is (r /|z|)d −2 .
Proof. Let B t be a d -dimensional Brownian motion, and let τr denote the hitting time of the closed
ball of radius r centered at the origin. Since the function z 7→ |z|2−d is harmonic in Rd \ {0}, we find
using Theorem 2.2.4 that M t B |B t ∧τr |2−d is a martingale (after smoothing the singularity at the
origin 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) = x 2−d . The optional stopping
theorem applied to M t implies
φ(x) = EM 0 = EM τr ∧τs = P(τr < τs )φ(r ) + P(τr ≥ τs )φ(s),
φ(s) − φ(|z|)
. Taking s → ∞, we find P(τr < ∞) = (r /|z|)d −2 .
which gives P(τr < τs ) =
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 (B T ∧t )t ≥0 be a Brownian motion started at z ∈ D and stopped at the exit
time T = inf{t ≥ 0 : B t ∈ C \ D}. By the open mapping theorem [9], f (D) is an open set. Therefore,
if we consider the image of (B T ∧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 Figure 1).
f (z) = exp(3z/2)
Figure 1: A Brownian motion started in the unit disk and its image under the
analytic function z 7→ exp(3z/2).
For all z 0 ∈ D with f 0 (z 0 ) , 0, the Taylor series development
f (z) = z 0 + f 0 (z 0 )(z − z 0 ) + O(|z − z 0 |2 ),
shows that f transforms points near z 0 by scaling z − z 0 by a factor of | f 0 (z 0 )| and rotating z − z 0
by an angle arg f 0 (z 0 ). Since Brownian motion is invariant under rotations and scaling (the latter
with a time change), we expect that f (B t )t ≥0 can be transformed into a Brownian motion with a
suitable time change. (The restriction that f 0 (z 0 ) , 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].
Theorem 2.3.1. (Itô’s formula) Let D be a domain in Rd +1 , let f be a twice continuously differentible function from D to R, and let M 1 , . . . , M d be continuous local martingales. Then up to the exit
time of (t , M t1 , . . . , M td ) from D, we have
Z t
∂f
1
d
1
d
f (t , M t , . . . , M t ) = f (0, M 0 , . . . , M 0 ) +
(s, M s1 , . . . , M sd ) d s
∂t
0
d Z t ∂f
X
+
(s, M s1 , . . . , M sd ) dM si
∂x
i
i =1 0
Z
d
d
1 X X t ∂2 f
+
(s, M s1 , . . . , M sd ) d [M i , M j ],
2 i =1 j =1 0 ∂x i ∂x 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 M 0 = 0 almost 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 M σ(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 (B t )t ≥0 be
a Brownian motion started at z ∈ D. If f : D → C is analytic, then there exists a Brownian motion
B̃ t in f (D) started at f (z) for which f (B t ) = B̃ R t | f 0 (B s )|2 d s .
0
Proof. Write z = x +i y, f (z) = u(x, y)+i v(x, y), and B t = X t +i Y t where X t and Y t are independent
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
d u(X t , Y t ) =
(X t , Y t ) d X t +
(X t , Y t ) d Y t
∂x
∂y
µ
¶
1 ∂2 u ∂2 y
1 ∂2 u
+
+
d
t
+
(X t , Y t ) d [X , Y ]t
2 ∂x 2 ∂y 2
2 ∂x∂y
∂u
∂u
=
(X t , Y t ) d X t +
(X t , Y t ) d Y t .
∂x
∂y
Similarly, d v(X t , Y t ) = ∂v/∂x(X t , Y t ) d X t +∂v/∂y(X
t . From these expressions we may comR t t ,0Y t ) d Y
2
pute the quadratic variation
[u(B
)]
=
[v(B
)]
=
|
f
(B
)|
d
s and the covariation [u(B ), v(B )]t = 0.
t
t
s
0
Rs 0
2
Let σ(t ) = inf{s ≥ 0 : 0 | f (B u )| d u > t }, and define B̃ t = f (B σ(t ) ) = u(B σ(t ) ) + i v(B σ(t ) ). By the
Dubins-Schwarz theorem, B̃ 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(z 0 ) =
h(z + r exp(i θ)) d θ.
0
Proof. Let τ be the exit time of a planar Brownian motion started at z 0 from the disk of radius r
centred at z 0 . By Theorem 2.2.4, (h(B t ))t ≥0 is a martingale. Apply the optional stopping theorem
R 2π
to find that h(z 0 ) = Ph(B τ ) = 0 h(z + r exp(i θ)) d θ, as desired.
Theorem 2.4.2. (Maximum Modulus Principle) If U ⊂ C is a domain and f : U → C is a nonconstant analytic function, then | f | has no local maxima in U .
Proof. Suppose for the sake of contradiction that there exists z 0 ∈ U and ² > 0 for which | f (z 0 )| ≥
| f (z 0 + r exp(i θ)| for all 0 < r < ². By adding a suitable constant to f , we may assume that the
image of {z : |z − z 0 | ≤ ²} under f is contained in the right half-plane. Therefore, log f is analytic
on {z : |z − z 0 | < ²}, from which it follows that the real part log | f | is harmonic. By the previous
theorem, for any 0 < r < ² we have that log | f (z 0 )| is an average of the values log | f (z 0 + r exp(i θ)|.
Therefore, log | f | is constant on ∆ B {z : |z − z 0 | < 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 t be a Brownian motion started at the origin. By Theorem
2.2.4, Re f (B t ) is a martingale. Since Re f is bounded, the martingale convergence theorem implies
that Re f (B t ) converges almost surely as t → ∞. On the other hand, Re f (C) contains more than
one element, since f is nonconstant. Choose α < β so that inf Re f (C) < α < β < sup Re 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 lim inf Re f (B t ) < α < β < lim sup Re f (B t ), almost surely. This contradicts the convergence of
Re f (B t ).
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) = e i θ z for all z ∈ D.
Proof. Let z 0 ∈ D and choose 0 < r < 1 so that z 0 is contained in the open disk centred at the origin
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 B t be a Brownian motion started at z 0 , and applying the optional
stopping theorem to g (B t ) to find
g (z 0 ) = Eg (B S )
Since | f (z)| ≤ 1 for all z ∈ D, we have |g (B S )| = | f (B S )|/|B S | ≤ 1/r . Letting r → 1 gives |g (z 0 )| ≤ 1.
Moreover, if |g (z 0 )| = 1, then z 0 is a local maximum for g . By the previous theorem, this implies
that g is constant, and |g | = 1. Therefore, there exists θ ∈ R for which g (z) = exp(i θ), so f (z) =
z exp(i θ).
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 constant.
Proof. Let B t be a Brownian motion in C started at the origin. By the previous theorem, f (B t ) is a
time change of a Brownian motion. Since Brownian motion visits every neighbourhood
of ∞, the
R∞
boundedness of f requires that that the time change does not go to ∞. That is, 0 | f 0 (B s )|2 d 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 n and Tn to be the nth entrance and
exit times, respectively, of D. Then S n and Tn are finite for all n almost surely. By the Markov
property of Brownian motion, the variables Tn − S n are i.i.d.
R ∞ Also, each hasPfinite 2expectation by
Proposition 2.1.12. By the strong law of large numbers, 0 | f 0 (B s )|2 d x ≥ ∞
n=1 δ (Tn − S 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
10
2 BROWNIAN MOTION AND COMPLEX ANALYSIS
question of the Hausdorff dimension of the Brownian frontier. We now present two simpler examples.
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 B t be a planar Brownian motion started at (², 0). Denote by n ² the number of
windings of B t around the origin before the first time B t hits the unit circle centred at the origin.
For all 0 < ² < 1, the law of 2πn ² / log ² is the Cauchy distribution.
Proof. Let X t be a Brownian motion started
at
¡
¢ (log ², 0) and stopped at τ = inf{t ≥ 0 : Re X t ≥ 0}.
Then B t is equal in distribution to exp X σ(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 2πn ² , since Im X t is a continuous realisation of the multi-valued function arg exp X t which starts at 0 (see Figure 2).
(log ², 0)
0
Xt
z 7→ exp(z)
(², 0)
0
Bt
Figure 2: The Brownian motion X t started at (log ², 0) and its image under the
exponential map.
Define x = − log ² and S t = sup0≤s≤t (Re X s + x), and observe that τ < t if and only if S t < x. Combining Propositions 2.1.5(ii) and 2.1.11, we find that the random time τ has the same distribution
as x 2 /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 τ = τ1/2 Im X 1 (to see that
this equality holds for all stopping times independent of X , note that it is straightforward
³
´ for a
d ¡x¢
Im X 1
−1
stopping times of the form n dτ/ne, and take n → ∞). Thus Im X τ = Z (Im X 1 ) = x Z . Recalling that the quotient of two independent standard normal random variables has the Cauchy
d
d
distribution, we have Im X τ = C x, where C is a Cauchy random variable. Therefore, 2πn ² = C x ⇒
d
2πn ² / log ² = C .
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 τ ∈ S) > 0.
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 conformal invariance, P(B τ ∈ S) = P(B τ0 0 ∈ ϕ(S)). Since ϕ¯∂D : ∂D → {z : |z| = 1} is a homeomorphism,
ϕ(S) has positive arc length. By rotational symmetry, P(B τ0 0 ∈ 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 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 Figure 5 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
¡
¢
P B [0, TR ] ∪ B 0 [0, TR0 ] does not disconnect the unit circle from ∞ = R −ξ(2,0)+o(1) ,
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 t = X t + i |Wt |. A Brownian excursion from 0 to ∞ in H is a process which is equal in
distribution to (E t )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),
and suppose that the exponential martingale Z t B exp(M t − 12 [M ]t ) of M is uniformly integrable
and that M 0 = 0 almost surely. Define a measure P̃ by d P̃/d P = Z∞ . If X is a continuous local
martingale under P, then X − [X , M ] is a continuous local martingale under P̃.
Theorem 3.2.2. Suppose σ : R → R and b : R → R are Lipschitz. If X and X̃ satisfy the stochastic
differential equation
dX t = σ(X t ) dB t + b(σt ) dt ,
(3.2.1)
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
12
then X and X̃ have the same distribution.
Proposition 3.2.3. For all z = x + i y ∈ H and R > y, a Brownian motion (Z t )t ≥0 = (X t + i Y t )t ≥0
started at z and conditioned to hit R + i R before R is equal in distribution to (Wt + i |W̃t |)t ≥0 , where
W and W̃ are independent standard Brownian motions started at 0 in R and at (y, 0, 0) in R3 , respectively.
Proof. For r ≥ 0, let τr denote the hitting time of R + i r . Define M t = y −1 Yτ0 ∧τR ∧t , and observe
that M t is a bounded nonnegative martingale with M ∞ = limt →∞ M t = Ry 1{τ0 >τr } . We define a new
measure P̃ by d P̃/d P = M ∞ , where P is the law of (B t )t ≥0 . Under P̃, we have
R tτ0 >−1τR almost surely.
To apply Girsanov’s theorem, note that M t is the exponential martingale of 0 M t dM t . Therefore,
the processes (X t )t ≥0 and
B t = Yt −
t
Z
0
M s−1 d [Y , M ]s
are continuous local martingales under P̃. Since [X , X ]t = [B, B ]t = t and [X , B ] = 0, X and B are
independent P-Brownian motions by Lévy’s characterisation. Moreover, under P̃,
Z t
ds
Yt = y + B t +
.
0 Ys
Using Itô’s formula with f (x) = |x| in the domain R3 \ {0}, we find
Z
3 Z t W̃ i
X
1 t 2ds
s
i
dW̃s +
|W̃t | = |W̃o | +
2 0 |W̃t |
i =1 0 |W̃ s |
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 (|W̃t |)t ≥0 satisfies the same stochastic differential
equation (3.2.1) as Y t under P̃, with σ = 1 and b(x) = 1/x. Although b is not Lipschitz, we may
replace b with a Lipschitz function b n which is equal to b up to time τ1/n . By Theorem 3.2.2, this
implies that for all n, (Y t )t ≥0 and (|W̃t |)t ≥0 have the same law up to time τ1/n . Taking n → ∞, we
find that (Y t )t ≥0 and (|W̃t |)t ≥0 have the same law up to time τ0 . Moreover, the law of Y t under P
conditioned on the event τR < τ0 is the same as the law of Y t under P̃, because Proposition 2.1.12
gives
P̃(Z t ∈ A) = E(M ∞ 1{Zt ∈A} ) =
P(Z t ∈ A and τ0 > τR )
R
P(Z t ∈ A and τ0 > τR ) =
,
y
P(τ0 > τR )
for all A ∈ B(C ([0, ∞), C). The last quotient is the elementary conditional probability of A given
that Z t hits R + i R before R.
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 K : H \ K → H which satisfies g K (z) − z → 0 as |z| → ∞.
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 g K . We use the notation X [a, b) for the trajectory {X t : a ≤ t < b} of a process (X t )t ≥0 on the
time interval [a, b).
Theorem 3.2.6. Let (E t )t ≥0 be a Brownian excursion and let K ∈ Q± . We have
P(E [0, ∞) ∩ K = ;) = g K0 (0).
Proof. First observe that by the Schwarz reflection principle, g K 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, g K0 (0) exists. By the continuity of g
on N ∩ H, we have g (N ∩ R) ⊂ R. This implies that g K0 (0) is real, which in turn implies g K0 (0) =
limw→0 Im g K (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 E t . By
the previous proposition,
P(B hits R + i R before R ∪ K )
P(E [t , τR ) ∩ K = ;) =
.
(3.2.2)
P(B hits R + i R before R)
A Brownian motion B started at z ∈ H has probability Im(z)/R of hitting R+i R before R, by Proposition 2.1.12. The probability that B hits R+i R before R∪K is equal to the probability that a Brownian
motion started at g K (z) hits g K (R + i R) before R, by conformal invariance. Since g K (z) − z → 0 as
z → ∞, we may choose R large enough that g K (R + i R) lies in the strip {z : R − 1 < Im z < R + 1}.
Therefore for R > 1,
Im g K (z)
Im g K (z)
< P(B hits R + i R before R ∪ K ) <
R +1
R −1
Substitute into (3.2.2) and take R → ∞ to find
µ
¶
Im g K (E t )
P(E [t , ∞) ∩ K = ;) = E
.
Im E t
Taking t → 0 and applying dominated convergence on the right-hand side gives P(E [0, ∞) ∩ K =
;) = g K0 (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) B
{r e i ϕ : r > 0 and 0 < ϕ < π − θ}. We will define a new process B̂ = X̂ + i Ŷ taking values in W (θ) by
reflecting B horizontally off the ray {r e i (π−θ) : r > 0} and stopping it when it hits the real line. More
precisely, let c = − cot θ, let Ŷ t = Y t , and let
µ
¶
X̂ t = X t + sup (cY t − X t ) ∨ 0 ,
s≤t
which is the unique process for which X̂ t ≥ cY t and `t = X̂ t − X t satisfies
(i) `t is nondecreasing and continuous, and
(ii) The Stieltjes measure d `t is supported on the set {t ≥ 0 : X̂ t = cY t }.
See [10] for a proof of uniqueness. The relationship between X t , cY t , and X̂ t is illustrated in Figure
3 below with toy functions X t and cY t .
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 (B̂ t )0≤t ≤τR+ be a reflected Brownian motion started at
z ∈ W (θ). Then (Φ(B̂ t ))0≤t ≤τR+ is a time change of a reflected Brownian motion started at Φ(z).
Proof. Define the ray ρ = {r e i (π−θ) : 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
X̂ t
cY t
Xt
Figure 3: The reflected process ( X̂ t )t ≥0 .
that the derivative exists because the boundary of W (θ) Ris straight in a neighbourhood of z ∈ ρ).
t
Also, since d `t is supported on {B̂ t ∈ ρ}, the process `˜t = 0 Φ0 (B s ) d`s is continuous, nondecreasing, and has its Stieltjes measure d `˜t supported on {Φ(B̂ ) ∈ ρ}. Therefore, it suffices to show that
Φ(B̂ ) − `˜ 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
( X̂ t , Y t ) d X̂ t +
( X̂ t , Y t ) d Y t − d `˜t
du( X̂ , Y ) =
∂x
∂y
∂u
∂u
=
( X̂ t , Y t ) (d X t + d `t ) +
( X̂ t , Y t ) d Y t − d `˜t
∂x
∂y
∂u
∂u
( X̂ t , Y t ) d X t +
( X̂ t , Y t ) d Y t ,
=
∂x
∂y
so the real part of Φ(B̂ )− `˜ is a local martingale. It follows from Lévy’s characterisation that Φ(B̂ )− `˜
is a time change of Brownian motion, as in the proof of Theorem 2.3.3.
W (θ)
H
b θ (z) = z π/(θ−π)
θ
0
0
Figure 4: A sample Brownian excursion reflected off the ray {r e 3πi /8 : r ≥ 0} in
W (θ) and its image under b θ .
Observe that that b θ (z) = z π/(π−θ) is a conformal map of W (θ) onto H which fixes the origin.
Definition 3.3.2. Let E be a Brownian excursion and let Ê be the reflection of E in the wedge W (θ).
A reflected Brownian excursion with angle θ is a process Ẽ which is equal in law to b θ (Ê ).
Theorem 3.3.3. Let Ẽ be a reflected Brownian excursion with angle θ. Then for all K ∈ Q+ , we have
P(Ẽ [0, ∞) ∩ K = ;) = Φ0K (0)1−θ/π ,
where ΦK (z) = g K (z) − g K (0) is the unique conformal isomorphism of H \ K → H which satisfies
ΦK (0) = 0 and ΦK (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
15
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
excursion started at Ẽ τr and stopped at τR has the same law as a reflected Brownian motion started
at Ẽ τr , conditioned on the set {τr < τ0 } and stopped at τR . Therefore, by Proposition 3.3.1, we may
compute as in the proof of Proposition 3.2.6,
·
¸
Im Φ̃(Ẽ τr )
P(Ẽ [τR , ∞) ∩ K = ;) = E
,
(3.3.1)
Im Ẽ τr
for all r > 0, where Φ̃ is the unique conformal map from W (θ)\b −1 (K ) → W (θ) that fixes the origin
and satisfies Φ̃(z)/z → 1 as z → ∞. Since b θ−1 ◦ΦK ◦b θ satisfies these requirements, Φ̃ = b θ−1 ◦ΦK ◦b θ .
As z → 0, we have
Φ̃(z) = b θ−1 (ΦK (b θ (z)))
³
´1−θ/π
π/(θ−π)
= ΦK (z
)
³
´1−θ/π
= Φ0K (0)z π/(θ−π) + O(|z 2π/(θ−π) |)
,
so Φ̃0 (0) = Φ0K (0)1−θ/π . Therefore, taking the limit as r → 0 in (3.3.1) and applying dominated convergence, we find
P(Ẽ [0, ∞) ∩ K = ;) = Φ0K (0)1−π/θ .
Let us make note of the case θ = 3π/8.
Corollary 3.3.4. If K ∈ Q+ , and Ẽ is a reflected Brownian excursion with angle θ = 3π/8, then
P(Ẽ [0, ∞) ∩ K = ;) = Φ0K (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 ∞ 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 g K be the unique conformal map of H \ K
onto H for which g K (z) − z → 0 as |z| → ∞. Then there exists a unique a K ∈ R for which g K (z) =
z + a K /z + O(|z|−2 ) as |z| → ∞. The quantity a K is denoted hcap(K ) and is called the half-plane
capacity of K . Define rad(K ) to be the radius of the smallest circle containing K whose centre is in
R. Let (K t )t ≥0 be an increasing family of compact H-hulls, and define K t ,t +h = g K t (K t +h \ K h ). The
family (K t )t ≥0 is said to satisfy the local growth property if K t ,t +h → 0 as h → 0, uniformly as t ranges
T
over compact subsets of R+ . If (K t )t ≥0 has the local growth property, then for all t , h>0 K t ,t +h
contains a single point, which we will define to be ξt . The function ξ : R+ → R is called the Loewner
transform of (K t )t ≥0 , and it is continuous. Loewner’s theorem describes a correspondence between
ξ and the differential equation
∂
2
g t (z) =
,
g 0 (z) = z,
∂t
g t (z) − ξt
which is called Loewner’s equation. Observe that the solution of Loewner’s equation does not necessarily exist for all t ≥ 0, since the denominator can go to zero. For each z ∈ C, we define T z to be
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)
16
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
0
κ = 8/3
κ=4
0
κ=6
0
Figure 5: Sample paths of numerical approximations of SLEκ for κ = 8/3, 4, and
6.
(i) Suppose that (K t )t ≥0 is an increasing family of compact H-hulls satisfying the local growth
property. Suppose further that (K t )t ≥0 has been parametrised so that hcap(K t ) = 2t . Then
for each z ∈ H, the solution of the Loewner equation up to time T z is given by t 7→ g K t (z).
(ii) Let (ξt )t ≥0 be a continuous, real-valued process. For z ∈ H, define g t (z) to be the solution of
Loewner’s equation up to time T z , and define K t = {z : T z ≤ t }. Then (K t )t ≥0 is an increasing
family of compact H-hulls satisfying the local growth property, hcap(K t ) = 2t , the Loewner
transform of (K t )t ≥0 is ξ, and g K t = g t for all t ≥ 0.
In the context of part (ii) of Loewner’s theorem, ξ is called the driving function. The SchrammLoewner evolution is obtained by taking the driving function of Loewner’s equation to be a Browp
nian motion with a diffusivity κ ≥ 0, i.e. ξt = κB t , where B t is a standard Brownian motion. It
can be shown [5] that the complement K t of H t 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 /c 2 )t ≥0 is also an SLEκ .
Proof. For each t ≥ 0, let g t be the conformal map corresponding to γt . Define ĝ t (z) = cg t /c 2 (z/c),
which is the family of conformal maps corresponding to cγt /c 2 . Then
∂ĝ t (z) 1
2/c
2
= ġ t /c 2 (z/c) =
=
,
p
p
∂t
c
g t /c 2 (z/c) − κB t /c 2 ĝ t (z) − κcB t /c 2
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 (K t )t ≥0 be a family of compact H-hulls satisfying the local growth property, let ξt =
T
h>0 K t ,t +h be the driving function associated with K t , and let (g t )t ≥0 be the corresponding family
of conformal maps g t : H \K t → H. Let Φ(z) = a 0 +a 1 z +a 2 z 2 +· · · be a conformal map defined in an
H-neighbourhood U of ξ0 . Note that for Φ(U ) ⊂ H, we must have a 0 , a 1 , a 2 , . . . ∈ R and a 1 > 0. Let
T = sup{t ≥ 0 : K t ⊂ U }. For 0 ≤ t < T , define K t∗ = Φ(K t ), a t∗ = hcap(K t∗ ), and let g t∗ be the unique
conformal map H \ K t∗ → H with g t∗ (z) − z → 0 as z → ∞. Finally, let Φt = g t−1 ◦ Φ ◦ g t∗ .
Proposition 3.5.1. With the preceding notation, we have
(i) (K t∗ )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 (K t∗ )0≤t <T ,
(iii) t 7→ a t∗ has derivative ȧ t∗ = 2Φ0t (ξt )2 , and
(iv) (t , z) 7→ Φt (z) is differentiable in both coordinates with
Φ̇t (ξt ) = −3Φ00t (ξt ),
Φ̇0t (ξt ) =
Φ00t (ξt )2
2Φ0t (ξt )
and
(3.5.1)
4
− Φ000
(ξt ).
3 t
(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 + Φ0t (ξt ))(g t (K t +h − K t ) − ξt ) = Φ0t (ξt )2 hcap(g t (K t +h − K t ) = 2hΦ0t (ξt )2 . To derive
(iv), assume first that t = 0. Write f t for g t−1 . Differentiate f t (g t (z)) = z with respect to t to find that
f˙t (z) =
Also, note that ġ 0∗ (z) =
ȧ t∗
z−ξ∗0
−2 f t0 (z)
z −U t
.
from Loewner’s equation, and apply the chain rule:
Φ̇0 (z) = ġ 0∗ (Φ0 ( f 0 (z))) + (g 0∗ )0 (Φ0 ( f 0 (z)))Φ00 ( f 0 (z)) f˙0 (z)
=
2Φ00 (ξ0 )2
Φ0 (z) − Φ0 (ξ0 )
2Φ00 (ξ0 )2
− (g 0∗ )0 (Φ0 ( f 0 (z)))Φ00 ( f 0 (z)) f 00 (z)
2Φ00 (z)
2
z −U0
−
.
Φ0 (z) − Φ0 (ξ0 ) z − ξ0
For 0 < t < T , we can apply the map g t to (K s )t ≤s<T and g t∗ to (K s∗ )t ≤s<T to reduce to the case t = 0.
Therefore
2Φ0t (z)
2Φ0t (ξt )2
−
.
(3.5.3)
Φ̇t (z) =
Φt (z) − Φt (ξt ) z − ξt
=
Letting z → ξt gives (3.5.1). Since the RHS of 3.5.3 is continuous, we can find Φ̇0 (z) by differentiating with respect to z to find
−2Φ0t (ξt )2 Φ0t (z)
2Φ00t (z)
2Φt (z)
+
−
,
(Φt (z) − Φt (ξt ))2 (z −U t )2
z − ξt
which gives (3.5.2) by letting z → ξt .
Φ̇0t (z) =
Proposition 3.5.2. The process M t = 1{t <τ A } Φ0t (ξt )α satisfies
·µ
¶
µ
¶
¸
(α − 1)κ + 1 Φ00t (ξt )2
κ 4 Φ000
d M t Φ00t (ξt ) p
t (ξt )
=
κ dB t +
+
−
dt.
αM t Φ0t (ξt )
2
Φ0t (ξt )2
2 3 Φ0t (ξt )
(3.5.4)
(3.5.5)
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
18
Proof. We apply Itô’s formula to find
·
¸
Φ00t (ξt ) p
Φ00t (ξt )2 Φ000
d M t Φ̇0t (ξt )
1
t (ξt )
=
dt + 0
κ dB t +
(α − 1) 0
+
κ dt ,
αM t Φ0t (ξt )
Φt (ξt )
2
Φt (ξt )2 Φ0t (ξt )
Equation (3.5.5) follows by substituting the expression in (3.5.2) for Φ̇0t (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 = ;) = Φ0K (0)5/8 ,
where ΦK (z) = g K (z) − g K (0) is the unique conformal isomorphism of H \ K → H which satisfies
ΦK (0) = 0 and ΦK (z)/z → 1 as z → ∞.
Proof sketch. Let D = H \ K , and for ² > 0, let D ² denote the set of all points in H whose distance
from K is at least ². Define g t , g t∗ , and Φt as in Proposition 3.5.1 with Φ = ΦD ² . Let τ be the hitting
time of R ∪ K and M t = 1{t <τ} Φ0t (ξt )α . Observe that choice κ = 8/3 and α = 5/8 makes (M t )t ≥0 a
local martingale, by Proposition 3.5.2. Since 0 ≤ Φ0t (ξt ) ≤ 1, (M t )t ≥0 is a martingale. Note that by
Proposition 3.2.6,
Φ0t (ξt ) = P(E [0, ∞) ⊂ g t (D ² )),
(3.5.6)
where E is a Brownian excursion started at ξt . Our goal is to show that M t approaches the indicator
of {τ = ∞} = {γ[0, ∞)∩K = ;} as t → ∞. Note that as t → ∞, g t (H \D ² ) gets smaller and farther from
the origin, which (when made rigorous) implies M t → 1 on the set {τ = ∞}. Also, on {τ < ∞}, γτ
lies on a ball B at least half of which is contained in H \ D ² . Therefore, there exists p > 0 for which
a Brownian motion started at any point in B ∩ H has probability greater than p of exiting H \ γ[0, t ]
on the boundary (−∞, γτ ) and also probability greater than p of exiting on (γτ , ∞). It follows that
g τ (B ∩ H) is contained in a proper cone with vertex ξτ . By (3.5.6), this implies Φ0τ (ξτ ) = 0. Thus
Φ0D ² (0)5/8 = M 0 = EM τ = 1 · P(γ ⊂ D ² ) + 0 · P(γ ∩ H \ D ² , ;) = P(γ ⊂ D ² ).
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
X
[
α
α
H² (A) = inf
(diam D n ) : A ⊂ D n ; ∀n, diam D n ≤ ² .
As ² → 0, the set of coverings
exists in [0, ∞].
n=1
{D n }∞
n=1
n
decreases, so H²α (A) increases. Therefore, lim²&0 H²α (A)
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
β
H² (A) = inf
= inf
½
∞
X
n=1
½∞
X
n=1
β−α
≤²
(diam D n ) : A ⊂
(diam D n )
inf
½
∞
X
n=1
β−α
=²
β
β−α
[
n
¾
D n ; ∀n, diam D n ≤ ²
α
(diam D n ) : A ⊂
α
(diam D n ) : A ⊂
H²α (A) → 0
[
n
[
n
¾
D n ; ∀n, diam D n ≤ ²
¾
D n ; ∀n, diam D n ≤ ²
as ² → 0.
Proposition 3.6.1 and establishes the existence of α0 such that H α (A) is equal to ∞ on the interval
(0, α0 ) and 0 on the interval (α0 , ∞). This critical value α0 is defined to be the Hausdorff dimension
of A, and is denoted dimH (A).
Lemma 3.6.2. H α (A) = 0 if and only if for all sequences ²m → 0+ and δm → 0+ , there exist sets Um, j
P
for which diamUm, j < ²m for all j and j (diamUm, j )α < δm .
Proof. Since H²α (A) increases as ² decreases, H α (A) = 0 if and only if H²α (A) = 0 for all ² > 0. The
equivalence of
H²α (A) = 0 for all ² > 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 all
z, w ∈ C), then dimH ( f (A)) ≤ b −1 dimH (A).
Proof. (From [5]) Let α > dimH (A), let ²m → 0+ and δm → 0+ , and let Um, j satisfy the conditions in
P
Lemma 3.6.2. Then the sets Ũm, j = f (V ∩ Um, j ) satisfy diam Ũm, j ≤ c²bm and j (diam Ũm, j )α/b ≤
P
c α/b j (diamUm, j )α < c α/b δm . Thus dimH f (A) ≤ α/b for all α > dimH A, which implies dimH ( f (A)) ≤
b −1 dimH (A).
Proposition 3.6.4. If A =
∞
[
n=1
A n , then dimH A = supn dimH A n .
Proof. It is clear that dimH A ≥ supn dimH A n . Conversely, let α > supn dimH A n . Let Un,m, j satP
S
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 Ũm, j = n Un,m, j . By
P
P
the triangle inequality, diam Ũm, 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 /c 2 )t ≥0 has the
same 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 random variable dimH X [s, t ] has the same distribution as dimH X [0, ∞).
S
Proof. Write X (0, ∞) = q X [q s, q t ], where the union ranges over all rationals q > 0. By the scaling
hypothesis, the distribution of the dimension of X [q s, q t ] does not depend on q. By Proposition
3.6.4, the distribution of dimH X [q s, q t ] is equal to that of X (0, ∞).
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
20
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 {s n }∞
n=1 and {t n }n=1 . Then
∞
X
¡
¢
¡
¢
P dimH (γ[σ, τ] = d ) =
P dimH X [s m , t n ] = d and (σ, τ) = (s m , s n ))
=
m,n=1
∞
X
m,n=1
¡
¢
P (σ, τ) = (s m , t n ) = 1.
For general σ and τ, define σn B n −1 dnσe and τn B n −1 bτnc. Since X [σ, τ] =
result follows from Proposition 3.6.6.
S
n
X [σn , τn ], the
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 SLE 8/3 ), 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
p κ is trivially 2, since the path is space filling.
Also, SLE0 is given by the simple formula γ(t ) = 2i t , 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 SLEκ
is linear.
¡
¢
Theorem 3.7.1. The Hausdorff dimension of SLEκ is almost surely 2 ∧ 1 + κ8 .
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 →
0. It takes Θ(²−2 ) balls of radius ² to cover a given (bounded) region in R2 , and of these Θ(²−d )
are needed to cover K . Therefore, we expect that the probability that an ²-neighbourhood of z
intersects K should take the form ²2−d 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 SLEκ path γ started at a visits the ²-neighbourhood of ζ ∈ H. We will
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
p
2dt
g dt , a maps to a 0 = a + κ dB t , ζ maps to ζ0 = ζ + ζ−a
(from the Loewner differential equation),
0
0
−2
and ² maps to ² = ²|g dt (ζ)| = ² − 2² Re((ζ − a) )dt . To obtain the last expression, differentiate
2dt
ζ 7→ ζ + ζ−a
and observe that for small z, |1 − z| ≈ 1 − Re z. Recall that that g dt (γ) is an SLEκ started
0
at a . Therefore, an SLE κ started from a 0 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
!#
¡
¢
p
2dt
2dt
P (ζ, ζ, ², a) = E P ζ +
,ζ+
, ² − 2² Re (ζ − a)−2 dt , a + κ d B t
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 E d B t = 0 and E(dB t )2 = d t , and subtract P (ζ, ζ, ², a) from
both sides to obtain Ã
!
¡
¢
2 ∂
2 ∂ κ ∂2
∂
+
+
− 2 Re (ζ − a)−2 ²
P = 0.
ζ − a ∂ζ ζ − a ∂ζ 2 ∂a 2
∂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
2y
2x
∂
∂
2 ∂ 2 ∂
+
− 2
.
= 2
2
2
ζ ∂ζ ζ ∂ζ x + y ∂x x + y ∂y
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(x 2 − y 2 ) ∂
∂
∂
−
+
−
P =0
(3.7.1)
²
x 2 + y 2 ∂x x 2 + y 2 ∂y 2 ∂x 2 (x 2 + y 2 )2 ∂²
We expect that P scales as some power of ²/|ζ|, and by scale invarance we expect that its dependence on ζ is a function of the argument θ of ζ alone. It turns out that a power of sin θ works. Write
µ
¶2−d ³
´α
y2
²
p 2 2
as ²2−d y d −2−2β (x 2 + y 2 )β and substitute in (3.7.1) to obtain
x 2 +y 2
x +y
¡
¢
−y −2+d −2β (x 2 + y 2 )−2+β ²2−d (κ − 8 − 2κβ)βx 2 + (4d − 8 − κβ)y 2 = 0.
Solving the system
κ − 8 − 2κβ = 0,
gives d = 1 +
κ
,
8
as desired.
4d − 8 − κβ = 0
It is worth emphasizing that the ideas in this derivation are only sufficient to give an upper bound
on the dimension, since a priori the form ²2−d 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 dimH (SLE κ ) ≥ 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] 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 = {S K : K ∈ Q+ }, where S K B {F : F ∩ K = ;}. Observe that the law of
F is determined by the values of P(F ∩ K = ;) for K ∈ Q+ , since S K 1 ∩ S K 2 = S K 1 ∪K 2 shows that E is a
π-system. Denote by ∂r F the right boundary ∂(H \ F ) \ (0, ∞) of the filling F .
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
22
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 whose
centres has rational coordinates. Enumerate the elements U1 ,U2 , . . . of U. Let B ∈ U, and define
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 1 , P 2 , . . . of P. If B intersects ∂r F , then there exists P ∈ P for which P \ B
is connected and ∂r F does not intersect the closure of any of the other balls whose union is P [9].
Therefore,
E (B ) =
∞
[
n=1
E (P n ) \ E (P n \ B ),
which shows that E (B ) ∈ Σ+ .
Observe that every D ⊂ H is contained in an element of U with diameter at most 2 diam D. Therefore, if we define for A ⊂ C,
½∞
¾
∞
X
[
α
α
H̃ (A) = lim inf
diam(D n ) : A ⊂
D n , diam D n ≤ 1/k, D n ∈ U for all n ,
k→∞
−α
α
n=1
n=1
α
α
we have 2 H̃ (A) ≤ H (A) ≤ H̃ (A). In particular, H̃ α is nonzero if and only if H α is nonzero, so
dimH (A) = inf{α : H̃ α (A) = 0}. Writing
∞
X
diam(Un )α 1E (Un ) 1diamUn ≤1/k
H̃ α (∂r F ) = lim
k→∞ n=1
shows that (α,V ) 7→ H̃ α (V ) is measurable, and finally
½
¾
dimH (∂r F ) = inf+ α1{H α (∂r F )=0}
α∈Q
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 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 leftfilling 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, µ1 (S) = 1. By Corollary 3.3.4 and Theorem 3.5.3, we have
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 : A ∩ F = ;} : F ⊂ Ĉ is closed and Ĉ \ F is simply connected .
Here Ĉ denotes the extended complex plane C ∪ {∞}. Define the frontier fr A of a set A ⊂ Ĉ 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)
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 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 set
of all balls in Ĉ with rational radius and centre either
© at ∞ or
ªat a point with rational coordinates.
−1
(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 + r e i θ : r ≥ 0},
and define the translation σξ (z) = z − ξ. Define a(A, x) = x + i sup{Im z : Re z = x and z ∈ A} and
b(A, x) = x + i inf{Im z : Re z = x and z ∈ A}. Choose y large enough that σ−i y (A) ⊂ H, let ξ = x + i y,
and define
³
³
³
´
³
´´´
∂r,x A = A ∩ σ−1
◦
∂
◦
σ
A
∪
ρ
a(A,
x),
π/2
∪
ρ
b(A,
x),
−π/2
,
r
ξ
ξ
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 precisely, z ∈ fr A if and only if there exists a path γ : [0, 1] for which γ(0) = z, γ(1) is in a neighbourhood 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
of
¡ component
¢ ¡
¢ 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
³
³ 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 intersect 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 t 0 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 0 . By compactness, there exists
² > 0 for which the strip
{z : x < Re z < γ(t 0 − ²) and Im z > Im a(A, x)}
lies in C \ A. Define a new path γ∗ which is equal to γ up to time t 0 − ²/2, follows a vertical ray out
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
24
to a neighbourhood of ∞ in C \ A on the interval [t 0 − ²/2, t 0 + ²/2], and follows a circular arc down
to R+ on the interval [t 0 + ²/2, 1].
d
Lemma
3.8.7. Suppose that (K t )t ≥0 is a family of random compact subsets of C for which K t =
p
t K 1 for all t > 0. Let τ be a random time whose range is countable. Then dimH K τ = d almost
surely if and only if dimH K 1 = d almost surely.
Proof. Suppose that dimH K 1 = d almost surely. By the scaling hypothesis, the distribution of
dimH K t does not depend on t , so dimH K t = d almost surely for all t > 0. Enumerate the elements
in the range of τ as {t n }∞
n=1 . Then
µ∞
¶
[¡
¢
P(dimH K τ = d ) = P
{dimH K tn = d } ∩ {τ = t n }
n=1
=
=
∞
X
n=1
∞
X
n=1
¡
¢
P {dimH K tn = d } ∩ {τ = t n }
P(τ = t n ) = 1.
S
Conversely, suppose that dimH K τ = d almost surely. If Ω is written as a disjoint union n Ωn and
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 K 1 , d ) > 0, we have
∞
X
¡
¢
P(dimH K τ = d ) =
P {dimH K tn = d } ∩ {τ = t n }
<
n=1
∞
X
n=1
P(τ = t 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 X σ 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 (X t )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 X σ , let the neighbourhood rational random time be
the smallest.
Theorem 3.8.9. If B is a Brownian motion in C, then dimH fr B [0, 1] = 4/3 almost surely.
Proof. Let Ê be the image of a reflected Brownian excursion under the map b θ−1 (z) = z 5/8 from H to
the wedge W (3π/8). It is easy to verify that the right boundary of Ê [0, ∞) is b θ−1 (∂r (bθ(Ê [0, ∞)))).
Let γ̃(t ) be a parametrisation of ∂r Ê [0, ∞), which has the same law as b θ−1 ◦ γ where γ is an SLE8/3
by Theorems 3.3.3 and 3.5.3. Also, b θ−1 preserves Hausdorff dimension by Propositions 3.6.3 and
3.6.4. Therefore, dimH ∂r Ê [0, ∞) = 4/3 almost surely.
Define the stopping times σ = inf{t ≥ 0 : Re Ê t = 2}, τ = inf{t ≥ σ : Re Ê t = 1}. By Proposition 2.1.8,
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
r
τ
25
σ1
σ2
θ
0
Figure 7: The labels σ1 , σ2 , and τ are displayed near the points Ê σ1 , Ê σ2 , and Ê τ
for a linear interpolation of a sample path Ê [0, τ]
σ and τ are finite almost surely. Let σ1 and σ2 be random times defined by
σ1 = inf{σ ≤ t ≤ τ : Ê t = sup{Im Ê s : Re Ê s = 2 and σ ≤ s ≤ τ}}
σ2 = inf{σ ≤ t ≤ τ : Ê t = inf{Im Ê s : Re Ê s = 2 and σ ≤ s ≤ τ}}
Let r = inf{ |Ê τ − Ê t | : t is between σ1 and σ2 }, and let τ0 be the neighbourhood rational random
time associated with τ and r . (Note that r > 0 by compactness). By the continuity of Ê ,
∂r,1 Ê [σ, τ] = ∂r,1 Ê [σ, τ0 ].
(3.8.1)
(1, n)
∆(Ê τ , n −1 )
θ
Figure 8: The shaded region represents the domain Un (z, θ).
Proposition 2.1.9 shows that dimH ∂r,2 Ê [σ, τ] ≤ d intersects (g ◦ γ)(I ) on an interval I of positive
length, which implies dimH ∂r,2 Ê [σ, τ] ≥ 4/3 almost surely by Proposition 3.6.7. Suppose that the
event C B {dimH ∂r,2 Ê [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 Un (z, θ) (see
Figure 8) by
Un (z, θ) = ∆(Ê τ , 1/n) ∪ (W (θ) \ ([1, ∞) × [0, n])) .
Define the events D n = {Ê [τ, ∞) ⊂ Un (Ê τ , θ))}. By Propositions 2.5.3 and 2.2.6, P(D n ) > 0 for all
n ∈ N. By the monotone convergence theorem, P(C ∩{Ê (σ, τ)∩Un (Ê σ , θ) = ;}) > 0 for large enough
3 SCHRAMM-LOEWNER EVOLUTIONS AND THE DIMENSION OF THE BROWNIAN FRONTIER
26
n. By the strong Markov property, C and D n are independent. Therefore, P(C ∩ D n ) > 0. Moreover,
we may cover ∂r,2 Ê (σ, τ] with countably many open disks which do not intersect Ê σ . If ∂r,2 Ê (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,2 Ê [σ, τ] has dimension greater than 4/3. With positive
probability, Ê [0, σ] does not intersect ∆. Since Ê [0, σ] is independent of (Ê σ+t − Ê σ )t ≥0 , we find
that ∆ ∩ ∂r,2 Ê [σ, τ] coincides with (b θ−1 ◦ γ)[0, ∞) on an interval of positive length, with positive
probability. Again by Proposition 3.6.7, this is a contradiction. Therefore, dimH ∂r,1 Ê [σ, τ] = 4/3
almost surely. From (3.8.1), we also have dimH ∂r,1 Ê [σ, τ0 ] = 4/3 almost surely.
Since Ê [σ, τ0 ] has the same law as a Brownian excursion started at Ê σ , up to time τ0 , we conclude
by Lemma 3.8.7 (along with a suitable translation) that dimH ∂r,0 E [0, T ] = 4/3 almost surely for a
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
B̃[0, T ∧T 0 ], where B̃ is a Brownian motion started at i and conditioned to hit R +2i before R. Thus
dimH ∂r,0 B̃ [0, T ∧ T 0 ] = 4/3 almost surely. Since B̃ is a conditioned Brownian motion, we have
¡
¢
¡
¢
1 = P dimH ∂r,0 B̃ [0, T ∧ T 0 ] = 4/3 = 2 P dimH ∂r,0 B [0, T ∧ T 0 ] = 4/3 and B hits R + 2i before R ,
which by symmetry implies P(dimH ∂r,0 B [0, T ∧ T 0 ] = 4/3) = 1. By Lemma 3.8.7, dimH ∂r,0 B [0, 1] =
4/3 almost surely. By Proposition 3.8.6 and symmetry, dimH fr B [0, 1] = 4/3 almost surely.
27
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