University of Copenhagen
Master’s Thesis in Statistics
The Schramm-Loewner Evolution
Mads Bonde Raad
supervised byprofessor Bergfinnur Durhuus and professor Jan Philip Solovej
April 15, 2016
1
Disposition
The purpose of this thesis is to construct the Schramm Loewner Evolution (SLE) trace, anddescribe some of its properties. Our aim will be to lead a discussion based on nothing butbasic complex analysis and stochastic calculus, with an exception of a few auxillary advancedresults in analysis and probability. During our discussion, we will treat the following fourtopics.
Compact H hulls and mapping-out functionsWe will introduce a certain kind of subset K ⊂ H called a compact H-hull. We shall provethe correspondance between compact hulls and a certain type of conformal H → H mapcalled mapping-out functions, and prove that these have Laurent series at z = ∞ given asg (z) = z+ az−1 +O
(|z|−2
). We will prove various properties of the mapping-out functions,
ultimately ending up in discussing Loewner flows. These are families of mapping-out func-tions that are induced by a certain kind of family of increasing compact hulls referred to asfamilies with the local growth property (LGP).
The Loewner differential equationWe aim to prove that the Loewner flow solve the Loewner differential equation for somedriving function ξ. This leads to the reverse discussion; starting with a driving functionξ, we prove that there is a unique solution to the Loewner equation. We aim to presentmany properties concerning these solutions, among other that they form a Loewner flow forsome LGP family. After establishing a confident base in the Loewner equation, and Loewnerflows in general, we introduce the probability aspect. We discuss the Loewner flow gt as adistribution induced by solving the Loewner equation whose driving function is a Brownianmotion with volatility parameter σ
2= κ.
SLEκ as a continuous process in the planeWe aim to prove that almost surely, g−1t extends continuously to the real line, and hencedefines the SLEκ trace; a curve in H given by γt = g−1t (ξ (t)) . Finally we aim to discussphases of the SLE. We will present the characteristics of the trace, and prove parts of it.
Simulation of the SLEAs a final goal, we shall attempt to present methods of simulating the SLE as well as pro-viding actual trajectories of the SLE trace.
2
Introduction
Colloquially speaking, The SLE may be described as a distribution on a system of infor-mation. The system is a soft word for four abstract mathematical elements, each able togenerate the remaining three. In this paper we shall discuss each of the four ways of describ-ing the SLE, and describe the relation to the other elements.
The first chapters are devoted to establishing theory concerning families of compact hulls inthe complex-halfplane H. A compact hull Kt is a bounded set in H, whose complement inH is open and simply connected. The SLE on compact hulls is a distribution on a nicely-behaved subset of compact-hull families, having the so-called local growth property, denotedLGP families in short. Any increasing family of compact hulls induces two new elements.The first is a driving function, which is a continuous function ξ : [0,∞) → R. The SLE asa distribution of ξ’s is simply a Brownian motion with a variance parameter traditionallydenoted as κ. The second induced element is a Loewner flow (gt) made of what we shall callmapping-out functions. These are conformal maps from Ht := H\Kt to H, made unique byensuring that gt ≈ z for large moduli. It turns out that ξ allows inversion; it generates thecorresponding Loewner flow and in turn the LGP family. The first is retrived as the uniquesolution to the Loewner differential equation given by
∂tgt (z) =2
gt (z)− ξ (t), g0 (z) = z.
The second is realized to be Kt = {z ∈ H : t ≥ ζ (z)} where ζ (z) denotes the maximal life-time of the solution to the Loewner equation and bears the name swallowing time of z.Even though there exist examples in the literature of inverse flows
(g−1t)
that do not extendcontinuously to z ∈ R, it is almost surely the case under the SLE law. This is the maintopic of the latter chapters, which are of a substantially more technical character. The SLEtrace may then be defined as γt = g
−1t (ξ (t)). For all κ > 0 the trace will be a continuous
[0,∞) → H function, but most properties of the process differ significantly depending onthe volatility parameter (in contrast to a Brownian motion). However, the trace does satisfythat we may retrieve Kt as all bounded path-components of H\γ [0, t] , so that also the tracecarries full information of the system.
The four objects and the described relations may be summarized in the following diagram.The numbers refer to the theorems that establish the relation.
γ6.1
-(Kt)t≥0
(gt)t≥0
6.1
6
�4.5
4.6
-
�
2.2,
4.4
ξ
4.3
?
3
Here follows a general overview of the thesis: In section 1 we develop a technique outsidethe field of classic conformal mapping theory that nevertheless shall help us in that field ata later point. That tool is the conformal invariance of a Brownian motion. The main resultis theorem 1.3.
In section 2 we present the compact hull and the mapping-out functions gK . The maintheorem is theorem 2.2.
Section 3 is a more technical section, where the goal is to build up estimates regarding thebehavior of gK . The absolute core estimate is theorem 3.5, but the more difficult theorem 3.7is also needed.
In section 4 we introduce the families of compact hulls, and the regularity condition”LGP”. We then prove that such families induce a continuous function ξ. We also reversethe process by starting with a continuous function ξ and deduce that the correspondingLoewner equation has a solution. We establish several properties that it must possess. Thenwe relate it to the setup that we discussed in the previous chapters, where we started offwith a family of compact hulls. This is done in theorem 4.6, which is the main result.
Section 5 is devoted to implement stochasticity into the Loewner setup, thereby obtainingthe Schramm-Loewner flow. The section mostly consists of definitions and explaining.
In section 6 we establish the existence of the trace γ. This is by far the hardest resultpresented in the dissertation, and the proof consists of several deep results, e.g. theorem 6.4,theorem 6.12, and lemma 6.13.
Section 7 is devoted to the SLE phases. The first subsection considers the Bessel flow inwhich we present the standard results within the field of stochastic calculus : lemma 7.3 andlemma 7.4. However, the main results are found in the end of the next subsection, where weuse the Bessel results to establish some properties of γ.
Finally we reach section 8 discussing simulation of the SLE. This section is meant asa more soft section. Instead of having main results, the section is more based on overalldescriptions of the algorithms presented etc.
In the end, the reader will find a fairly big appendix. Many, although not all, of thetheorems that I refer to may be found there.
Apart from mathematics on undergraduate level, the reader is assumed to be familiar withstochastic integration with respect to continuous semimartingales and measure-theoreticconditioning theory. Our main references for these are [Sok14] and [NH14]. Graduate levelcomplex analysis including boundary behavior of conformal maps will also prove as an ad-vantage.
All chapters, with the exception of 6 and 8, are primarily based on the [NB14]. Chapter 6is based on the original article by Oded Schramm and Steffen Rohde [SR05], and chapter 8is based on Tom Kennedys articles [Ken08],[Ken09].
4
A warm thanks goes to my supervisors Bergfinnur Durhuus and Jan Philip Solovej for super-vising me. I also thank Aske Thorn Iversen for fruitful discussions. Finally, my gratitudesgoes to Martin Emil Jakobsen for engaging persistently and energized in mathematical dis-cussions of any sort and size.
Contents 5
Contents
1 Conformal Invariance of a Brownian Motion 6
2 Compact H Hulls and the Mapping-Out Function 14
3 Estimates for gK 20
4 The Loewner Transform 284.1 Families of Hulls and the Local Growth Property . . . . . . . . . . . . . . . 284.2 Inversion of the Loewner Equation . . . . . . . . . . . . . . . . . . . . . . . 31
5 The Schramm-Loewner Flow 38
6 Existence of the Schramm-Loewner Evolution 416.1 Proof of the SLE Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7 Properties and Phases of the SLEκ 647.1 The Bessel Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.2 From the Bessel Process To SLEκ Phases . . . . . . . . . . . . . . . . . . . . 697.3 Proofs of the Weak Characterizations . . . . . . . . . . . . . . . . . . . . . . 75
8 Simulation of SLEκ 79
9 Final Words 85
10 Appendix 8610.1 Holomorphic, Univalent and Conformal Maps . . . . . . . . . . . . . . . . . 8610.2 Martin Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8910.3 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.4 Various Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.4.1 Topology and Connection . . . . . . . . . . . . . . . . . . . . . . . . 9410.4.2 ODE theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11 Notation 98
Conformal Invariance of a Brownian Motion 6
1 Conformal Invariance of a Brownian Motion
We start in an area that at first glance may seem auxiliary to the theory relevant for thisdissertation. It turns out that Brownian motions on complex domains are closely connectedto harmonic, and hence also holomorphic, maps. Knowledge of how a Brownian motion willhit the boundary of a domain, shall help us to achieve several results in the fields of complexanalysis and conformal maps. The main result in this chapter is that for arbitrary domains(not necessarily simply connected) with a conformal map between them, a Brownian motionis mapped to a Brownian motion, at the cost of a time-change. The theorem is very similarto the Dubins-Schwarz Theorem, but we present a proof for completeness. The very firstlemma in this thesis hints that Dubins-Schwarz is closely connected to our result, statingthat the conformal map of a Brownian motion is a local martingale. Before reading on, weurge the reader to consult the section of the appendix that discusses probability theory. Hereyou will find various notation and results which we will frequently make use of. See also theappendix concerning complex analysis and conformal maps.
Let B be a Brownian motion on C started at some z. Define (Ft)t≥0 as the completionof σ (Bs)s≤t satisfying the usual hypothesis. This is our default filtration, and all adaptedprocesses mentioned in the text to come, are adapted to Ft unless otherwise mentioned.
Lemma 1.1 (Ito’s formula for harmonic maps).Let D,K be domains in C. Take Z = (X, Y ) ∈ cS(D) and assume that [X] = [Y ] and thatX and Y has evanescent covariation.
1. If u : D → R is a harmonic map it holds up to indistinguishability (u.t.i) that
u(Z) = u(X0, Y0) + ∂1u (X, Y ) ·X + ∂2u (X, Y ) · Y
In particular u (Z) ∈ cMl
2. If φ ∈ H (D,K) , it holds that φ (X + iY ) ∈ cMl (K)
◦
Proof.Indeed, by A. 12 it is sufficient to show that the finite variation part of U (Z) given by
1
2
(∫ t0
∂11u (Xs, Ys) d[X]s +
∫ t0
∂22u (Xs, Ys) d[Y ]s
)+
∫ t0
∂12 (Xs, Ys) d[X, Y ]s
is evanescent. The latter term is evanescent since the integrator is assumed to be evanes-cent. The variation processes are equal so the first two terms are just the integral of theLaplace-operator, which is 0 since u is harmonic.
The second statement follows immediately, since φ = u + iv where u, v are harmonicmaps.
Conformal Invariance of a Brownian Motion 7
In order to make the thesis, and in particular this section, readable, we shall not writeu.t.i. at every inequality. Instead, i will remind the reader from time to time that a null-setof trajectories may differ at virtually all process inequalities in this section.
Before we prove conformal invariance of a Brownian motion, we present a lemma that willbe useful at several occasions.
Lemma 1.2.Let z0 ∈ D ⊂ C be a domain.
1. There exists a sequence of bounded domains (Dn) which satisfies that
z0 ∈ D1, Dn ⊂ Dn+1,⋃n∈N
Dn = D.
Let now X be a continuous process starting in z0, and let Tn be the first time X hits∂Dn. Let T be the first time X hits ∂D.
2. It holds that supn∈N Tn = T.
◦
Proof.Define f : D → R as the continuous map f (z) = d (z,Dc). We may assume that z0 = 0,and that f (0) > 1 after distorting D with a suitable linear map. Define
An =
{z : d (z,Dc) >
1
n
}∩B(0, n).
It is a sequence of increasing sets, and since An = f−1 ( 1
n,∞)∩ B(0, n) the sets are open.
Define Dn as the subset of An that are connected to 0. We see that 0 ∈ D1 and that Dn isopen as well. To prove the increasing-property of the domains, note that
Dn ⊂ An ⊂{z ∈ D : d (z,Dc) ≥ 1
n
}∩B(0, n) ⊂ An+1.
So it will suffice to show that ∂Dn is connected to 0. For z ∈ ∂Dn we may choose a ballB (z, r) ⊂ An+1. This ball must intersect Dn at a point w which is connected to 0 by a pathin Dn. Hence that path, together with the segment [z, w], form a path in An+1 from z to0, proving the desired. We claim that Dn ↑ D. To show that, fix any z ∈ D and choosea path γ ⊂ D from z0 to z (we abuse the notation and let γ denote the parameterizationand the image). Since γ is compact, f(γ) attains a minimum in some point p ∈ γ, andsupw∈γ |w| < N for some N ∈ N. Since p is an interior point in D, it holds that f(p) > 1/Nby choosing N large. It follows that γ ⊂ AN and therefore also z ∈ DN which proves theclaim.
Set R = supnTn. The second claim is that R = T . In the event R = ∞ the claim is
Conformal Invariance of a Brownian Motion 8
trivial. If R < ∞ occurs the claim follows from showing that XR ∈ Dc =⋂k∈N
Dck, which
we shall do in the following. If we consider a fixed k ∈ N it holds for all n ≥ k thatXTn ∈ Dcn ⊂ Dck. Indeed Dck is closed, so we can take the limit over n and obtain XR ∈ Dck.As k was chosen arbitrary, we conclude XR ∈ Dc which is the desired result.
We now prove the main result.
Theorem 1.3 (Conformal Invariance of a Brownian Motion).Let D,K be domains in C and φ : D → K be a conformal map. Set B,B to be Brownianmotions, started at z ∈ D and w = c+ di = φ (z) , respectively. Let T, S be the exit times
T = inf (t ≥ 0 : Bt /∈ D) S = inf (t ≥ 0 : Bt /∈ K) .
Define also
It =
∫ t0
|φ′ (Bs)|2 ds, T̃ = IT , τ (t) = inf (s ≥ 0 : Is = t) .
Define Wt = φ(Bτ(t)
)for t < T̃ . It holds that W has a limit for t ↑ T̃ , and with Wt := WT̃−
for t ≥ T̃ it holds that (T̃ ,W
)D=(S,BS
). (1)
◦
Proof.First, consider the case where D is bounded, and φ has a C1 extension to D. Write B =X + iY , where X, Y are independent Brownian motions started at z. In this case we notethat T < ∞ almost surely, due to neighborhood recurrence of B. Define the processes Zand A by
Zt = φ(BTt)
+Bt −BTt , At = ITt + t− T ∧ t.
The idea of the proof is to apply the Levy Characterization theorem ( A. 13) on Z aftera suitable time-distortion, to achieve that Z is indeed a time-changed Brownian motion.Therefore, the analysis following below suits to discover properties of the coordinate pro-cesses of Z, along with properties of their covariation function. However, we shall start witha discussion of A, and derive what will become our time-distorter τ .
Since φ is conformal, φ′ does not vanish. Hence A is strictly increasing on t < T . Fort ≥ T we have At = T̃ + t− T , so A is strictly increasing at all times. we conclude that A isa self-homeomorphism on [0,∞). Evidently the inverse of A coinsides with τ for t ≤ T̃ , andhence i denote the inverse of A as τ as well.
Conformal Invariance of a Brownian Motion 9
We now consider the coordinate processes of φ, given by φ (x+ iy) = u (x, y) + iv (x, y)and realize that Z = M + iN where
M = u(XT , Y T
)+X −XT , N = v
(XT , Y T
)+ Y − Y T .
We aim to prove that u(XT , Y T
)∈ cMl
(D), but to apply Ito’s formula, we need an open
event-space on which u is C2. Instead, take Dn, Jn as given in lemma 1.2 with respect tothe process X + iY and the domain D. Then XJn + iY Jn ∈ cS (D), and we may applylemma 1.1 to see that u
(XJn , Y Jn
)∈ cMl and
u(XJn , Y Jn
)= c+ ∂1u
(XJn , Y Jn
)·XJn + ∂2u
(XJn , Y Jn
)· Y Jn . (2)
Continuity of u on D gives that u(XJnt , Y
Jnt
)converges pointwise for n → ∞ to a limit in
D. Hence the right-hand side of (2) also have an a.s. limit. To determine the a.s. limit,it will suffice to find the p-limit. We find such one by applying dominated convergence forstochastic integrals theorem 2.3.1 [Sok14]. To satisfy the assumptions of this theorem, Definethe processes Hni , Hi for n ∈ N, i = 1, 2 by
Hni,t = 1J0,JnK (t) ∂iu(XJnt , Y
Jnt
), Hi,t = 1J0,T J (t) ∂iu
(XTt , Y
Tt
).
It is immediate that almost surely
Hni,s → Hi,s ∀s ≤ t.
pointwise for i = 1, 2 and for each fixed t ≥ 0. Note also that t 7→ |Hi,t| is an stochastic-integrable upper bound since ∂iu is continuous on D , i = 1, 2. We can now apply dominatedconvergence on each process to conclude that the following holds, atleast for each fixed t
u(XJnt , Y
Jnt
)− c = (Hn1 ·X)t + (H
n2 · Y )t
P→ (H1 ·X)t + (H2 · Y )t .
It follows from uniqueness of limits that almost surely
u(XTt , Y
Tt
)− c = (H1 ·X)t + (H2 · Y )t , ∀t ∈ Q+.
In that event, we have two continuous trajectories that agrees on t dense in [0,∞), and hencethey agree everywhere. From the definition of the stochastic integral, we see that it makesno difference whether we have 1J0,T J or 1J0,T K in the integrand. Hence we have obtained thefollowing equality up to indistinguishability
u(XT , Y T
)− c =
(∂1u
(XT , Y T
)·XT
)+(∂2u
(XT , Y T
)· Y T
). (3)
Evidently, the analogous result holds for v. From the right side of (3) we see that u(XT , Y T
)∈
cMl, which makes M a sum of local-martingale processes so M ∈ cMl too. An analogousargument shows that v
(XT , Y T
), N ∈ cMl, and finally we conclude that Z ∈ cMl (C).
Using standard calculation-rules for quadratic covariation, we obtain
[M ]t = [u(XT , Y T
)+X −XT ]t
= [u(XT , Y T
)]t + [X −XT ]t + 2
[u(XT , Y T
)T,(X −XT
)T︸ ︷︷ ︸=0
]t. (4)
Conformal Invariance of a Brownian Motion 10
The second term reduces to [X]t +[XT]t− 2
[X,XT
]t
= t− t∧ T . We insert (3) in the firstterm and apply usual rules for covariation processes:
=[∂1u
(XT , Y T
)·XT
]t+[∂2u
(XT , Y T
)· Y T
]t+ 2
[∂1u
(XT , Y T
)·XT , ∂2u
(XT , Y T
)· Y T
]t
=
∫ t∧T0
(∂1u (Xs, Ys))2 ds+
∫ t∧T0
(∂2u (Xs, Ys))2 ds+ 2∂1u
(XT , Y T
)∂2u
(XT , Y T
)·[XT , Y T
]t︸ ︷︷ ︸
=0 u.t.i
= ITt .
Inserting back into (4) gives that [M ] = A. Analogous calculations shows that [N ] = A aswell. Per definition of the variation process, M2 − A ∈ cMl and N2 − A ∈ cMl. We nowturn our attention to the covariation process [M,N ].
[M,N ] = [u(XT , Y T
)+X −XT , v
(XT , Y T
)+ Y − Y T ].
After using bi-linearity of covariation, all but one term trivially vanish. The term remain-ing are
[u(XT , Y T
), v(XT , Y T
)]. We evaluate it using (3) and another application of
bi-linearity.[u(XT , Y T
), v(XT , Y T
)]=[∂1u
(XT , Y T
)·XT , ∂1v
(XT , Y T
)·XT
]+[∂1u
(XT , Y T
)·XT , ∂2v
(XT , Y T
)· Y T
]+[∂2u
(XT , Y T
)· Y T , ∂1v
(XT , Y T
)·XT
]+[∂2u
(XT , Y T
)· Y T , ∂2v
(XT , Y T
)· Y T
].
Basic rules again applies. The second and third term in the sum above will have integrator0, and is therefore evanescent. The sum of the first and 4th term will be óne integral withrespect to
[XT]
=[Y T]
and with integrand
∂1u∂1v + ∂2u∂2v = ∂2v (−∂2u) + ∂2u∂2v = 0,
due to the Cauchy Riemann equations. Thus the entire covariation process is evanescent,which happens if and only if MN ∈ cMl.
Define the filtration Gs = Fτ(s), and the new processes M̃, Ñ , Z̃ by M̃s = Mτ(s), Ñs = Nτ(s)and Z̃s = Zτ(s). A. 14 gives that these are all cMl (Gs) processes. Furthermore [M̃ ] = [Ñ ] =s 7→ s and [M̃, Ñ ] is evanescent, since M̃Ñ ∈ cMl (Gs) . It follows from Levy characterizationtheorem that Z̃ = M̃ + iÑ is a Brownian motion on C. Note that
φ(B[0, τ
(T̃)) (ω)
)= φ (B[0, T ) (ω)) ⊂ K.
Recall that T
Conformal Invariance of a Brownian Motion 11
as well. This proves that T̃a.s.= inf
{t ≥ 0 : Z̃t /∈ K
}. Define the measurable maps
F :C ([0,∞) ,C)→ [0,∞],G :C ([0,∞) ,C)→ [0,∞]× C ([0,∞) ,C)
F (f) = {inf t ≥ 0 : f(t) /∈ K}G(f)(t) = (F (f) , f(min(t, F (f) ))).
(5)
We obtain (T̃ ,W T̃
)u.t.i= G(Z̃)
D= G(B) u.t.i=
(S,BS
)which is the desired result.
For the general result, take a sequence of domains (Dn) ↑ D as given in lemma 1.2, anddefine Kn = φ (Dn). We introduce Tn, Sn as the first times B, B leaves Dn, Kn, respectively.Note that Dn ⊂ Dn+1 gives that φ is C1 on Dn and hence satisfies the assumptions on thespecial case. Lemma 1.2 2) gives that Tn ↑ T and Sn ↑ S. Moreover we observe that T̃n = ITnhas upwards limit T̃ by monotone convergence (the limits may be∞). We show that Wt hasa limit for t ↑ T̃ whenever T̃
Conformal Invariance of a Brownian Motion 12
we may now use our distribution equalities(T̃n∨j,W
T̃n∨j)
D=(Sn∨j,BSn∨j
)to conclude that
= limN,n,...,j
P⋂
q1,q2∈Q,|q1−q2|
Conformal Invariance of a Brownian Motion 13
φ
Figure 1: A plot illustrating D being mapped conformally by φ to D =(0, 1)2 \
⋃n∈N [2
−n, 2−n + i/2]. The arc segments marked with red are ”mapped” to the linesegments as shown. The arc segments must be smaller and smaller, and thus they have anaccumulation point ( the red dot). This rough sketch illustrates how the blue curve in D,converging to the accumulation point, will be mapped to a curve that ends up oscillatingnear [0, i/2].
However, without even knowing D, we now know that a conformal map between D andD carries trajectories of a Brownian motion, to curves with limits in ∂D. So colloquiallyspeaking, any curve in D that are mapped through φ to a oscillating curve, must differ innature from trajectories of a Brownian motion. This is more of an indication, that an actualresult, but we shall later see how conformal invariance will help us to obtain several resultswithin the field of conformal maps.
Compact H Hulls and the Mapping-Out Function 14
2 Compact H Hulls and the Mapping-Out FunctionOur homebase for the SLE will be the open upper half-plane H. We start by defining thecompact hulls of H. A compact hull K induces a simply connected subdomain H ⊂ H.By the Riemann mapping theorem, H can be conformally identified with H by some mapφ. The first theorem in this chapter is a core theory within this dissertation, describing aspecific conformal map gK : H → H called the mapping-out function. The remaining of thischapter is devoted to prove basic properties of gK , and to introduce a nonnegative valueaK associated to K, called the half-plane capacity. The reader is strongly encouraged totake a look at the complex-analysis section of the appendix before reading on. In particularA. 1,A. 3, and A. 4 is important.
Definition 2.1 (Compact hulls).Let (K,H) be a partition of H. We say that K is a compact H-hull or just a compact hull,if K is bounded and H is a simply connected domain. Define also
Θ−K = inf{x ∈ R : x ∈ K
}, Θ+K = sup
{x ∈ R : x ∈ K
}.
◦
Note that it is immediate from the definition that K\K ⊂ R (the word compact is mis-leading, as K is not closed) and together with H0 (see appendix) it forms a partition of R.
In the proof for the theorem to come we need the automorphisms of the half plane. Themost important may very well be f : z 7→ −1
zwhich is an automorphism of H and −H.
Recall that a general map is a H automorphism if and only if it is a Möbius transformationwith real coefficients.
Theorem 2.2.Let K be a compact hull and H = H\K. There exists a unique map gK ∈ C (H) satisfyingthe hydronormalization condition
gK(z)− z → 0 for z →∞.
Moreover, gk satisfies the following properties
1. The map gK extends by reflection to a univalent g∗K : H
∗ → C map which also satisfiesthe hydronormalization condition.
2. The reflected mapping is strictly increasing on{x < Θ−K
}∪{x > Θ+K
}.
3. It holds that
gK (z)− z =aKz
+O(|z|−2
)(7)
for some aK ∈ R.
Compact H Hulls and the Mapping-Out Function 15
4. It holds that g∗K maps bounded sets to bounded sets, and that z 7→ g∗K (z)−z is bounded.
5. The inverse map (g∗K)−1 also maps∞ continuously to itself, and satisfies the hydronor-
malization condition.
◦
The reader is encouraged to see the first part of the appendix if not already familiarwith this topic. We shall use notation presented there, and also many results concerningconformal mappings.
Proof.By the Riemann mapping theorem there is a map φ1 ∈ C (H). Since K is bounded by someR > 0, AH (0, R,∞) ⊂ H. It follows that BH (0, 1/R) ⊂ f (H) , so applying A. 9 to φ1 ◦ fgives that φ1 extends continuously to ∞. After post-composing with a suitable Aut (H)map, we get a new map φ2 ∈ C (H) with φ2 (∞) = ∞. By definition of H0, there is a ballBH (x, r) ⊂ H for arbitrary x ∈ H0. Therefore another application of A. 9 gives that φ2extends by continuity to H0, and since φ2 (∞) = ∞, H0 must be mapped into R by theuniqueness part of A. 8. Now φ2 extends by Schwarz reflection to a univalent φ
∗2 : H
∗ → Cmap. We now establish a Laurent expansion of φ2 near ∞. We see that φ∗ = f ◦ φ∗2 ◦ fis univalent on f (H∗) which contains B′ (0, 1/R) and by Riemann’s theorem for removablesingularities, it extends analytically to z = 0 with φ∗ (0) = 0.
We now taylor expand φ∗ at z = 0 and achieve some m ≥ 1 such that
φ∗ (z) =∞∑n=m
anzn |z| < R−1
for an ∈ R (The coefficients are real due to reflection invariance). Indeed f (φ∗ (z)) will havea pole of order m at z = 0 and hence a Laurent series of the form
f (φ∗ (z)) =∞∑
n=−m
bnzn z ∈ B
(0, R−1
)where bn ∈ R. We now get a Laurent expansion for φ∗2 by inserting f (z)
φ∗2 (z) =m∑
n=−∞
cnzn |z| > R
where cn = (−1)n b−n ∈ R. We claim that m = 1 and c1 > 0. Indeed, assume for contradic-tion that m > 1. Assume that cm > 0, and consider z (t) = te
iθ for fixed θ ∈ (π/m, 2π/m)and t > R. The Laurent series gives that
|φ∗2 (z (t))− cmzm (t)||=cmzm (t)|
=1
|cm sin (mθ)||φ∗2 (z (t))− cmzm (t)|
tm≤ 1|cm sin (mθ)|
m−1∑n=−∞
|cn|∣∣tn−m∣∣
Compact H Hulls and the Mapping-Out Function 16
which vanishes for t → ∞. Therefore, there exist large t0 such that z (t0) ∈ H and|φ∗2 (t0)− cmzm (t0)| < |=cmzm (t0)|. But t 7→ cmzm (t) is a halfline in −H, and hence theinequality implies that φ2 (z (t0)) lies there as well, which contradicts with the image of φ
∗2.
For the case m ≥ 1, cm < 0 the argument may be repeated with θ ∈ (0, π/m).
Now, note that
g∗K (z) :=φ∗2 (z)− c0
c1
is also a univalent H∗ → C map which satisfies reflection, and whose restriction gK toH is a C (H) map. The map has the desired Laurent expansion, with aK = c−1/c1 ∈ Rand in particular it satisfies the hydronormalization condition for z ∈ H. This concludesthe existence part of the proof. Property 1) and 2) follows trivially, and we have alreadyproven 3). It remains to prove property 4), 5) and uniqueness. Point 4) is a straight forwardconsequence of φ∗ being homeomorphic around 0 with φ∗ (0) = 0 and that
gK (z) =f ◦ φ∗ ◦ f (z)− c0
c1.
To prove 5), note that for any sequence zn →∞, the sequence g−1K (zn) must be unbounded,implying that it has a subsequence converging to ∞. This proves the continuous exten-sion g−1K (∞) =∞. The hydronormalization property for the inverse follows immediately; ifzn →∞ then wn = g−1K (zn)→∞ and hence g
−1K (zn)− zn = wn − gK (wn)→ 0.
Finally we reach uniqueness. If h is another conformal H → H with the hydronormal-ization property, then Λ := h ◦ g−1K ∈ Aut (H) . This is the group of Möbius maps with realcoefficients. Note now that
Λ (z)− z = h(g−1K (z)
)− g−1K (z) + g
−1K (z)− z → 0 for z →∞.
This shows that Λ (∞) =∞ implying that Λ is in fact affine. The only affine map satisfyingthe hydronormalization condition is the identity, and the argument is completed.
Definition 2.3.For a compact hull K, the function gK, uniquely determined by the theorem above, will bereferred to as the mapping-out function. The real number aK will be referred to as the half-plane capacity of K and the map z 7→ gK (z) − z will be referred to as αK. Finally, denoteC±K as the limit of gK (x) for x→ Θ
±K from right and left, respectively. ◦
The hydronormalization condition says that gK acts on H by smoothening K down to thereal line. One of the main topics in this dissertation is regularity of g−1K on R. When can wepull K back up from the real line by g−1K ? We already know that it extends homeomorphicon the unbounded intervals
(−∞, C−K
)∪(C+K ,∞
)(marked with non-dashed red in the plot
below). We also know that in case K allows bounded intervals in H0t these induce intervalsin[C−K , C
+K
]that are mapped homeomorphic (marked with dashed-red).
Compact H Hulls and the Mapping-Out Function 17
gK
+ΘΘ C C+
Figure 2: A plot illustrating the concepts presented in the previous theorem. K is the unionof the two dashed areas on the left plot, and H0 is the red lines on the same plot
Theorem 2.4.Assume that K1 ⊂ K2 are compact hulls.1) For any r > 0, x ∈ R it holds that the mapping-out function for rK1 + x is given by
g (z) = rgK1
(z − xr
)+ x.
2) It holds that K1,2 := gK1 (K2\K1) is a compact hull and
gK2 = gK1,2 ◦ gK1 , aK2 = aK1,2 + aK1 .
◦
Definition 2.5.The compact hull K1,2 given in theorem 2.4 will be referred to as the remainder hull of K1w.r.t. K2 and gK1,2 as the remainder map. ◦
The names given do not appear in other literature, but we use these variables so muchthat they deserve a name.
Proof of theorem 2.4.The first statement is just a matter of verifying that g indeed is a mapping-out function forthe affine transformation of K1. I will leave it to the reader.
For the second statement we note that H2 = H\K2 and K2\K1 partitions H1. So as gK1maps H1 bijectively to H, and the latter set is mapped to K1,2, the first set must be mappedto H1,2 := H\K1,2. This proves that H1,2 is simply connected and since gK1 maps bounded
Compact H Hulls and the Mapping-Out Function 18
sets to bounded sets, K1,2 is a compact hull. Therefore gK1,2 and g := gK1,2 ◦ gK1 are well-defined conformal maps from respectively K1,2, K1 to H. We now verify that g satisfies thehydronormalization condition. With w := w (z) = gK1 (z), we know that w is eventuallywell-defined and going to ∞ for z → ∞ and therefore the Laurent expansion for the twomapping out functions gives that
z (g (z)− z) = z(αK1,2 (w) + αK1 (z)
)→ aK1,2 + aK1 for z →∞.
This proves the hydronormalization condition and additivity of the capacities.
Example 2.6.Let us discuss some specific compact hulls
1. The semi-disc K = D+ := D ∩ H is a compact hull, since H = H\D+ is simplyconnected. It is well known that the Joulowski map
gK (z) = z + z−1
is a conformal H → H map. Evidently zαK (z) = 1→ 1 for z →∞, so the hydronor-malization condition is satisfied. Therefore gK is indeed the mapping-out function forK and aK = 1.
2. Consider the vertical slit I = (0, i]. It is immediatly seen to be a compact hull aswell. The map z2 + 1 takes HI := H\I conformally to C\ [0,∞). With the squareroot√z given by z 7→ exp
(12Log2π z
)we conformally map that domain to H. So gI (z) =√
z2 + 1 is a conformal HI → H map. We see that
zαK′ (z) = z
(√z2 + 1− z
) (√z2 + 1 + z
)(√z2 + 1 + z
) = z√z2 + 1 + z
→ 12
for z →∞.
So gI is the mapping-out function of I with capacity aI =12.
From the construction of gI we also see that the Martin boundary can be identifiedwith the left- and right side of I, so that δI = I+ ∪ I−. The homeomorphic extensionof gI maps I
− to [−1, 0] and I+ to [0, 1].
◦
Note that both of the found capacities were positive. In general we have that the capacityof any nonempty hull is positive. For now, we prove a result giving a geometric/probabilisticintuition of the capacity, which additionally hands out non-negativity for free.
Theorem 2.7.Let K be a compact hull and assume that t 7→ Bt is a C-Brownian motion started at z ∈ H.Let T be the first time B leaves H and define the process G : t 7→ gK (Bt).
1. It holds that
E (GT− −BT ) = αK (z) .
Compact H Hulls and the Mapping-Out Function 19
2. The following characterization of the capacity is true
aK = limy→∞
yE= Biy,T .
◦
Proof.Recall the results of conformal invariance from the previous chapter. Since H is simplyconnected, T will be almost surely finite, meaning that B has a limit for t ↑ T . Conformalinvariance gives that gK
(Bτ(t)
)also has a limit for t ↑ T̃ proving that GT− is almost surely
well defined as well.
To prove the first claim, take domains Dn in H and hitting times Tn with respect to theprocess B as given in lemma 1.2. We apply lemma 1.1 to see that αK
(BTn
)∈ cMl (C),
and theorem 2.2 allows us to upgrade to αK(BTn
)∈ cMb (C). Optional stopping time with
times 0, Tn gives that EαK (BTn) = αK (z). After taking the limit n→∞, the claim followsby applying dominated convergence.
To prove the 2nd claim, let t 7→ Biy,t denote a Brownian motion started at iy. Recallthat Giy,T− ∈ R and hence =Giy,T− = 0. If we go into the above with z = iy we get
yE= Biy,T = −y=E (Giy,T− −Biy,T ) = −y=αK (iy)
We use the Laurent expansion to conclude that the right-hand side converge to aK fory →∞.
We will prove positiveness of the capacity in the next chapter. The result pops upautomatically while proving a somewhat deep theorem concerning an interaction of gK mapsfor two different hulls. The result just proven hints why it is not quite elementary that thecapacity is positive. This characterization extends the notion of capacity to bounded setsK ⊂ H such that H = H\K is a domain. But these sets does not necessarily have positivecapacity, see for example K = {i} .
Estimates for gK 20
3 Estimates for gK
The goal of this section is to prove some technical results that we will need in order toestablish the SLE, as a process of ”well-behaved” increasing families of compact hulls. Inparticular we shall need the so-called Continuity esimate theorem 3.5 and the differentia-bility estimate theorem 3.7. The section does not contribute to an overall understanding ofthe SLE, and can be skipped on a first-read. The reader is encouraged to take a look at thepart of the appendix that discuss the Martin Boundary.
We start by introducing the harmonic measure. If the reader is familiar with harmonicanalysis, he will surely already know the concept of such objects, but we will introduce themin a different, more probabilistic setup. Consider a simply connected domain D. Let B be aC-Brownian motion started at w ∈ D and let T be the first time B reach ∂D. As we saw inthe first chapter, T is almost surely finite. Therefore BT is well defined. Recall the Martinboundary, and the corresponding notation. By conformal invariance φ (Bt) also convergesalmost surely for t ↑ T for any φ ∈ C (D,D). Equivalently it almost surely holds that allsequences Btn (ω) , tn → T are dφ cauchy-sequences, and identified with the same point inδD. We call this point B̂T . The harmonic measure on
(δD, B̂
), denoted hwD, is defined as
B̂T (P ). When D is a Jordan domain, the Harmonic measure is simply the distribution ofBT on D. The first result states the harmonic measure on H and D. The Martin boundariesare immediately seen to be (identified with) R ∪ {∞} and the interval [0, 2π), respectively.In both cases the relevant conformal maps are well known Möbius transformations given by,say
φwH (z) =z − wz − w
, φwD (z) =z − w1− wz
.
Straight forward calculations now leads to the following result
Lemma 3.1.
1. Take w = x+ iy in H. It holds that hwH = f · λ where
f (t) =y
π((t− x)2 + y2
) .and λ is the Lebesque measure.
2. Let w ∈ D. It holds that hwD = g · λ where
g (t) =1− (x2 + y2)
2π((cos t− x)2 + (sin t− y)2
)1[0,2π) (t) .◦
Estimates for gK 21
We now present a result which will be surprisingly useful in producing estimates in com-plex analysis/conformal mappings. The theorem is equally important because it illustratesa type of argument concerning B̂T probabilities, which will be used many times throughoutthe dissertation. This particular argument will be more detailed than the ones to come, asthey are similar. Recall the Borel σ-algebra δBH on the Martin Boundary.Theorem 3.2.Let K be a compact hull, and let (Bx+iy,t)t≥0 be a Brownian motion started at a point x+iy ∈H and let T be the first time B leaves H. Let λ be the Lebesque measure. For any C ∈ δBHit holds that
limn→∞
πynP(B̂xn+iyn,Tn ∈ C
)= λ (gK (C))
for sequences zn = xn + iyn which satisfies yn →∞ , xn/yn → 0. ◦Proof.Write gk = u + iv. Define un, vn as the points u (xn, yn) , v (xn, yn). The imaginary part ofthe hydronormalization property, gives that
vn/yn → 1 for n→∞. (8)
If xn 6= 0 eventually, then un/xn → 1 as well and henceunyn
=unxn
xnyn→ 0. (9)
If xn = 0 infinitely often, then we may take a subsequence so that zn = iyn. The hydronor-malization property gives that
vn − iunyn
=gn (zn)
zn→ 1 for n→∞.
and hence (9) follows. All relevant sequences, has a subsequence that converges to the samelimit 0, which implies that the limit (9) is true for all relevant sequences. For fixed n, let Bbe the Brownian motion started at xn + iyn. Recall that gK , as any C (H) map, extends toa Ĥ → Ĥ homeomorphism by gK
((zn)n
)=(gK (zn)
)n.We have
πynP(B̂Tn ∈ C
)= πynP
(gK
(B̂Tn
)∈ gK (C)
).
Recall that δH is identified with R ∪ {∞} by (wn) ≡ limnwn for each (wn) ⊂ δH. Recallalso that G : s 7→ gK
(Bτ(s)
)is itself a Brownian motion started at un + ivn, and with a limit
in δH for s ↑ T̃n
Estimates for gK 22
where we have used lemma 3.1 point 2). The limits (8),(9) make the integrand converge to1gK(C) (t). If λ (gK (C)) < ∞ then 1gK(C) (t) maxn yn/vn is an integrable upper bound andwe apply dominated convergence. If λ (gK (C)) =∞ we apply Fatou. In both cases we get∫
gK(C)
vn/yn(t−unyn
)2+(vnyn
)2 dt→ λ (gK (C)) for n→∞which is the desired result.
It is particularly interesting at the unbounded parts of H0.
Corollary 3.3.If (a, b) ⊂ (−∞,Θ−) or (Θ+,∞) then
limn→∞
πynP (Bxn+iyn,Tn ∈ (a, b)) = gK (b)− gK (a) .
◦
We shall define the radius of a set K ⊂ C a bit differently than usually defined in theliterature. We insist on the ball covering K are centered in R. More precisely we define theradius of K as
R (K) = inf{r ≥ 0 | ∃x ∈ R : K ⊂ rD + x
}.
Lemma 3.4.Let K be a compact hull.
1. It holds that gK (x) ≥ x for x > Θ+. The opposite inequality will hold if x < Θ−.
2. If we add the assumptions K ⊂ D and x > 1 or x < −1 respectively, then we also havethe upper/lower bound gK (x) ≤ 1x + x or gK (x) ≥ x+
1x
respectively.
◦
Proof.For the first statement, let Biy be a C-Brownian motion started at iy for large y. Set τ, Sas the exit times of the sets H and H respectively. If Biy leaves H for the first time through(x, b), then it is certainly also the first time it leaves H and therefore
P (Biy,S ∈ (x, b)) ≤ P (Biy,τ ∈ (x, b)) ∀b > x.
The mapping-out function of H is the identity. So when we apply corollary 3.3 on both sidesof the inequality above we get
gK (b)− gK (x) ≤ b− x.
Estimates for gK 23
This implies that αK (b) + x ≤ gK (x) for all b sufficiently large. Taking the b → ∞ limitshows the desired.
To prove the second statement, define a third stopping time T as the first time Biy leavesH\D+. As before we have
P (Biy,T ∈ (x, b)) ≤ P (Biy,S ∈ (x, b)) ∀b > x.
We saw in example 2.6 that The mapping-out function of D+ is z + z−1, so when we applycorollary 3.3 on both sides of the above we get
b+1
b−(x+
1
x
)≤ gK (b)− gK (x) .
Equivalently
gK (x) +1
b≤ α (b) + x+ 1
x.
Taking the b → ∞ limit gives the desired result. The analogous procedure with b → −∞gives the reverse inequalities.
We already know that αK is bounded, from the construction of gK . It turns out that wecan estimate the bound with the size of K. For the proof of the coming theorem, let (b, a)for b > a denote the set (−∞, a) ∪ (b,∞).
Theorem 3.5 (The Continuity Estimate).For any compact hull K it holds that
|α∗ (z)| ≤ 3R (K) , z ∈ H∗.
◦
For two compact hulls K1 ⊂ K2 and z ∈ H2 we may enter this theorem with w = gK1 (z)to see that
gK2 (z)− gK1 (z) = g1,2 (w)− w = α1,2 (w) < R (K1,2) .
A peak at definition 4.1 shows the justification of the theorem’s name
Proof.We start by proving the result restricted to z ∈ H, and with K ⊂ D such that R (K) = 1.Recall from theorem 2.7 that if B is a Brownian motion starting at z ∈ H then αK (z) =E (GT −BT ). If ω ∈ Ω satisfies that |BT | > 1 then BT ∈ H0, and hence GT = gK (BT ) . Weapply lemma 3.4 and realize that
|αK (BT )| = |gK (BT )−BT | ≤1
|BT |< 1.
Estimates for gK 24
If |BT | ≤ 1, we claim that |GT | ≤ 2. Applying lemma 3.4 again we get that gK (x) ∈(x, x+ 1
x
)for any x > 1. For any ε > 0 we can find x0 close to 1 such that
x+1
x< x0 +
1
x0< 2 + ε ∀x < x0.
proving that gK (x) < 2 + ε for all x ∈ (1, x0). Note that Θ+ ≤ 1 so the restriction of gKto (1,∞) is strictly increasing by theorem 2.2. Therefore we conclude that gK ((1,∞)) =(c+,∞) for some c+ ∈ [1, 2]. A analogous argument proves that gK ((−∞,−1)) = (−∞, c−),where c− ∈ [−2,−1], and in total we see that gK maps the H0 interval (1,−1) homeomorphicto (c+, c−) ⊃ (2,−2) . Under the false assumption GT ∈ (2,−2), g−1K would send GT =gK (BT ) back to (1,−1), contradicting the assumption |BT | ≤ 1. So in this case it must holdthat |GT | ≤ 2. In any case, the triangle inequality gives
|GT −BT | ≤ 3.
which is the desired result.
For a general hull we may per definition of R (K), choose r = R (K) and some x ∈ Rso that K ′ = K−x
rsatisfies the assumptions under which we have just proved the result.
Therefore , we can obtain an estimate for gK , K = rK′ + x by theorem 2.4 as
|αK (z)| =∣∣∣∣rgK′ (z − xr
)+ x− z
∣∣∣∣ = r ∣∣∣∣αK′ (z − xr)∣∣∣∣ ≤ 3r.
The bound immediately extends to {z : z ∈ H} by reflection and also to H0 by continuityof g∗.
The next theorem shows that theO (|z−2|) term in the Laurent series for gK , can be chosenuniformly for all compact hulls of same size. First we establish some auxiliary estimate ona complex function.
Lemma 3.6.There is a constant C ∈ R with the following properties: For all θ ∈ [0, π] the inequality∣∣∣∣ 1z + z−1 − 2 cos θ − 1z
∣∣∣∣ < C |z|−2is true for |z| ≥ 3/2. ◦
Proof.After basic manipulations we arrive at the statement to show
∃c ∈ R+ ∀θ ∈ [0, π] , z /∈ B (0, 3/2) : c <∣∣∣∣z2 + 1− 2z cos θ2z2 cos θ − z
∣∣∣∣ .
Estimates for gK 25
We see that the numerator of the expression to the right has exactly two roots eiθ, e−iθ whichlie on the unit circle. We also have that∣∣∣∣z2 + 1− 2z cos θ2z2 cos θ − z
∣∣∣∣ ≥ |z2 + 1| − |2z|2 |z2|+ |z| → 12 for z →∞.From this convergence, we obtain a radius R so that the expression is greater than 1/4 for|z| > R. The annulus A (0, 3/2, R) is compact, and therefore we can find m > 0,M > 0 suchthat ∣∣z2 + 1− 2z cos θ∣∣ > m, ∣∣2z2 cos θ − z∣∣ < M for z ∈ A (0, 3/2, R) .using that both expressions are continuous as functions of (z, θ) . We have now justified thatc = max
(1/4, m
M
)serves as needed.
We now formulate and prove the differentiability estimate. First, the reader should befamiliar with elementary conditioning theory, found at for example [NS14]. This includesMarkov kernels.
Theorem 3.7 (The Differentiability Estimate).There exist C ∈ R with the following properties: For all r ∈ R+, x ∈ R and any compact hullK ⊂ rD + x the inequality ∣∣∣∣αK (z)− aKz − x
∣∣∣∣ ≤ CraK|z − x|2 ,holds for z such that |z − x| ≥ 2r. Additionally it is true that
aK = 0⇔ K = Ø.
◦
Proof.We prove for the special case r = 1 x = 0. The generalization is analogous to the one we didin the proof of the continuity estimate. Define D = H\D+, start a Brownian motion B atz ∈ D and consider the exit times T ≤ S
Estimates for gK 26
Then ψ (W,BT ) = (Ws +BT )s≥0. By the substitution theorem, found at [NH14] theorem2.1, we get that the conditional distribution (Qp)p∈C of (Ws +BT )s≥0 given BT = p
Qp = ψ (W, p) (P ) = (Ws + p)s≥0 (P ) .
This is the distribution of a Brownian motion starting at p and in particular it does notdepend on z. Now, notice that Bs+T = Ws + BT , and in particular BS = WS−T + BT . Itfollows that S − T is the first time Ws +BT leaves H. Therefore we can define
F : C [(0,∞) ,C)→ [0,∞] , F (f) = inf {t ≥ 0 : f (t) /∈ H}
and write =BS = F (ψ (W,BT )). The substitution theorem gives that the conditional distri-bution of =BS given BT = p is the Markov kernel F (Qp), which does not depend on z either.
With this fact in mind, we return to (10). Using properties of condition expectations we get
E(1(BT∈R)E (=BS|BT )
)= E
(1(BT∈R)=BS
)= E
(1(BT∈R)=BT
)= 0.
So 1(BT∈R)E (=BS|BT ) = 0 almost surely. Abstract change of variable gives that
E (E (=BS|BT )) =∫
(BT∈∂D+)E (=BS|BT ) dP (11)
=
∫(0,π)
E(=BS|BT = eiθ
)dArg BT (P ) (θ)
We shall now identify the sub-probability measure given as the restriction of Arg BT (P ) to∂D+. By example 2.6 we have that gD
(eiθ)
= eiθ + e−iθ = 2 cos θ. For any 0 < θ1 < θ2 < πwe apply that and conformal invariance of a Brownian motion to obtain
P (Arg BT ∈ (θ1, θ2)) = P (gD (BT ) ∈ (2 cos θ2, 2 cos θ1))= h
gD(z)H (2 cos θ2, 2 cos θ1)
=
∫ 2 cos θ12 cos θ2
1
π=(
1
t− gD (z)
)dt
=
∫ θ2θ1
1
π=(
2 sin θ
2 cos θ − gD (z)
)dθ.
so Arg BT (P ) restricted to ∂D+ has density w.r.t. λ(0,π) equal to the integrand above. Wenow combine (10),(11) and the density just found to conclude the integral representation of=αK given by
=αK (z) =∫ π
0
1
π=(
2 sin θ
gD (z)− 2 cos θ
)E(=BS|BT = eiθ
)dθ.
From this equation we see that if E(=BS|BT = eiθ
)= 0 λ(0,π) almost everywhere then
=gK (z) = =z for all z ∈ D. By uniqueness of holomorphic maps it follows that gK (z) = z
Estimates for gK 27
for all z ∈ H. That is the mapping-out function for K = Ø, in which case the claims aretrivial. In the other case, define b (θ) ∈ [0,∞) , a ∈ R+ and the map f ∈ H (H∗\ {0}) as
b (θ) =2 sin θ
πE(=BS|BT = eiθ
), a =
∫ π0
b (θ) dθ, f (z) = αK (z)−a
z.
Let C > 0 be the constant from lemma 3.6. With f = u+ iv we see that
|v (z)| =∣∣∣=αK (z)−= a
z
∣∣∣=
∣∣∣∣∫ π0
b (θ)=(
1
gD (z)− 2 cos θ
)dθ −=
(1
z
)∫ π0
b (θ) dθ
∣∣∣∣≤∫ π
0
b (θ)
∣∣∣∣ 1gD (z)− 2 cos θ − 1z∣∣∣∣ dθ < Ca|z2|
for any |z| ≥ 3/2. For z such that |z| ≥ 2 we can also get a bound on the derivatives asfollows. Define the domain Dz = A (0, 3/4 |z| ,∞). Note that d (z, ∂Dz) ≥ 14 |z|. We applyA. 11 to see that
|∂1v (z)| ≤4Ca
π |z|2· 4|z|≤ C
′a
|z|3.
Where C ′ is a new constant, not depending on z nor K. The same bound holds for |∂2v (z)|,and by the Cauchy Riemann equations, the bound also hold for |f ′ (z)|. We conclude that
|f (z)| ≤ C′a
|z|2for z ≥ |2| .
Which is the desired result, if we can show that a = aK . This is a consequence of limituniqueness; we have just showed that zf (z) → 0 for z → ∞, but the hydronormalizationcondition gives that zf (z) = z (αK (z)− a/z)→ aK − a.
The Loewner Transform 28
4 The Loewner Transform
In this section we shall introduce the Loewner differential equation. That will in turn inducea one-to-one correspondance between a continuous function ξ, a family of compact hulls(Kt), and a family of mapping-out functions (gt) solving the Loewner equation.
4.1 Families of Hulls and the Local Growth Property
From here on, we shall mostly deal with compact hulls as a part of a family(Kt)t≥0, and
in that setup we get rid of the double subscript, by introducing notation like gt,at and Rtfor the variables gKt , aKt and R (Kt) etcetera. We will also leave the ∗ notation for thereflected maps g∗, α∗. Instead we will write gt (z) even if =z ≤ 0. Recall the definition of theremainder hulls K1,2 from theorem 2.4.
Definition 4.1.Let (Kt)t≥0 be a family of compact hulls.
• We say that (Kt)t≥0 is increasing if Ks ( Kt for s < t.
• We say that (Kt)t≥0 has the Local Growth Property (referred to as the LGP) if (Kt)t≥0is increasing and
R (Ks,t)→ 0 for t ↓ s
uniformly on compact sets of s.◦
It would be deceiving to describe the LGP property as a growth-continuity condition, sinceit is possible, and quite natural in fact, to work with families of hulls that have discontinu-ously increasing area functions. It would be more precise to describe it as an insurance ofcollaboration between the mapping-out functions when smoothening out a compact hull. Ifwe first map a large indiced hull Kt through a smaller indiced mapping-out function gs, thenthe remaining Ks,t should be small if t− s is small.
Example 4.2.Recall example 2.6.The family (It)t≥0 given by It = (0, ti] and with mapping-out functions gt (z) =
√z2 + t2 is
an LGP family. For fixed t > 0 then It+h\It = (ti, (t+ h) i] which is mapped through gt to(0, hi], so Rt,t+h = h which converges 0 for t+ h→ t uniformly over t.
The family (Kt) given by Kt = tD+ and with mapping-out functions gt (z) = z + t2z−1is not an LGP family. Indeed, Kt+h\Kt contains the annulus AH (0, t, t+ h). A neighbour-hood around ±t in there is mapped through gt to neighbourhoods of ±2t, so the radius neverdecreases below 2t, declining convergence of 0. ◦
Let us extract some properties of LGP families.
The Loewner Transform 29
Theorem 4.3.Let (Kt)t≥0 be an LGP family.
1. It holds that Kt,t+ = Ø and in particular that Kt+ = Kt.
2. The map t 7→ at is continuous and strictly increasing.
3. For any t ≥ 0 there is a unique ξ (t) ∈ R such that ξ (t) ∈ Kt,t+h for all h > 0.Moreover ξ as a map on [0,∞) is continuous.
◦
Proof.Note that by definition or Rt, we have that =z > Rt ⇒ z /∈ Kt. For any fixed z ∈ H, t ≥ 0,the LGP gives some h > 0 so that =z > Rt,t+h. This proves that z /∈ Kt,t+h and thereforeKt,t+ is empty, implying that Kt = Kt+. For the second statement, recall from theorem 2.4and theorem 3.7 that as + as,t = at and as,t 6= 0 since Kt\Ks 6= Ø. It follows that t 7→ at isstrictly increasing, and it remains to prove that a is continuous. Take t0 > 0, t ∈ [0, t0) , h > 0and note that Kt,t+h ⊂ Rt,t+hD+ +x for some x, and that the compact hull Rt,t+hD+ +x hascapacity R2t,t+h. Monotonicity and remainder-additivity of the capacity gives that
|at+h − at| = at,t+h ≤ R2t,t+h. (12)
Rt,t+h converges uniformly to 0 on t ∈ [0, t0] and hence right continuity in t follows. To showleft continuity We use a uniformity argument, which will be used several times during thischapter. It goes as follows ; for ε > 0 we take δ > 0 so that Rs,s+h < ε for all s ∈ [0, t0] withh ∈ (0, δ). For all h < δ we take s = t− h and note that
|at−h − at| = |as − as+h| ≤ R2s,s+h < ε2.
This proves left-continuity. For the last claim, pick xn ∈ Kt,t+n−1 . Since these are decreasingcompact sets, we may assume, after choosing a subsequence, that xn converges to some limitξ (t) . For all m ∈ N (xn) lies in Kt,t+m−1 eventually, so ξ (t) lies there as well. On theother hand, the LGP ensures that there can be no more than one element in
⋂h>0Kt,t+h and
that such element must be real. We prove continuity of the ξ map. Fix t ≥ 0 and considerarbitrary h ∈ (0, 1) and z ∈ Kt+2h\Kt+h. We see that gt (z) ∈ Kt,t+2h and gt+h (z) ∈Kt+h,t+2h. Hence it must hold that
|ξ (t)− gt (z)| < 2Rt,t+2h, |ξ (t+ h)− gt+h (z)| < 2Rt+h,t+2h.
The continuity estimate gives that
|gt+h (z)− gt (z)| < 3Rt,t+h
and so we obtain an upper bound
|ξ (t)− ξ (t+ h)| ≤ 5Rt,t+2h + 2Rt+h,t+2h.
The LGP gives that Rs,r converges uniformly to 0 on s ∈ [0, t+ 1], and therefore rightcontinuity follows. The argument for left continuity is a uniformity argument analogous tothe one showing left continuity of t 7→ at.
The Loewner Transform 30
We say that ξ is the Loewner transform of(Kt)t≥0. The just proven theorem states
that A : [0,∞) 3 t 7→ at/2 is strictly increasing and thereby forms a homeomorphism from[0,∞) to [a0/2, a∞/2). A direct calculation shows that the LGP property is invariant overre-parameterizations, so if (Kt)t≥0 is an LGP family of hulls, then (K
′s)s∈[a0/2,a∞/2) given by
K ′s = KA−1(s) also has the LGP property. The half-plane capacity of K′s = KA−1(s) is a
′s = 2s.
There is a tradition for saying that this family is parameterized by half-plane capacity eventhough that is literally only half the truth. We are only going to be interested in familiesthat have a0 = 0, a∞ =∞, meaning that the family parametrized by capacity is also on theform (K ′t)t≥0 and satisfying K
′0 = Ø.
We now define the swallowing time of z ∈ H given by ζ (z) = inf {t ≥ 0 : z ∈ Kt}. Since Ktis right-continuous, it holds that z ∈ Kζ(z) and in particular ζ (z) > 0 for all z ∈ H. Wemay begin to harvest the fruits of the previous chapter, by treating gt (z) as a function oft ∈ [0, ζ (z)) also. It is immediate from the continuity estimate ( and the remark after ) thatt 7→ gt (z) is continuous for each fixed z. We now show how the differentiability estimateand continuity combined, will ensure differentiability of t 7→ gt (z).
Theorem 4.4.Let (Kt)t≥0 be a LGP family, parametrized by capacity. For any z ∈ H, it holds that[0, ζ (z)) 3 t 7→ gt (z) is differentiable and satisfies the Loewner equation
∂tgt (z) =2
gt (z)− ξ (t), g0 (z) = z.
Moreover, it holds that gt (z)− ξ (t)→ 0 for t ↑ ζ (z) whenever ζ (z) is finite. ◦
Proof.Take arbitrary fixed z ∈ H, t ≥ 0, and choose h0 > 0 such that t + h0 < ζ (z). To provedifferentiability, note that for s ≤ t, h < h0 we have that ξ (s) is a point in Ks,s+h bydefinition. Therefore it must hold that Ks,s+h ⊂ 2Rs,s+hD + ξ (s). We may therefore applytheorem 3.7 with the remainder hulls and get∣∣∣∣gs+h (z)− gs (z)h − 2gs (z)− ξ (s)
∣∣∣∣ = 1h∣∣∣∣αs,s+h (gs (z))− 2hgs (z)− ξ (s)
∣∣∣∣ ≤ 2C (2Rs,s+h)|gs (z)− ξ (s)|2(13)
for all h < h0, s ≤ t satisfying |gs (z)− ξ (s)| ≥ 4Rs,s+h. Indeed s 7→ |gs (z)− ξ (s)| is acontinuous positive map so if we restrict it to s ∈ [0, t+ h0], for h0 sufficiently small, it isbounded from below by some c > 0. By the LGP we may take h0 smaller, to ensure that|gs (z)− ξ (s)| > c > 4Rs,s+h holds true for all s ≤ t, h < h0. Hence we have∣∣∣∣gs+h (z)− gs (z)h − 2gs (z)− ξ (s)
∣∣∣∣ < 4CRs,s+h/c2, ∀s ≤ t, h < h0.Right- and left- differentiability at z may now be obtained from the LGP.
For the last claim, we note that for any s < ζ (z) we have z ∈ Kζ(z)\Ks and thereforegs (z) ∈ Ks,ζ(z). Indeed ξ (s) ∈ Ks,ζ(z) so we get |gs (z)− ξ (s)| ≤ 2Rs,ζ(z), and we can con-clude our proof by applying LGP.
The Loewner Transform 31
Notice that the limit denies any extension that solves the Loewner equation for longerthan ζ (z). Note also that Ht, Kt are given by the swallowing times as
Ht = {z ∈ H : t < ζ (z)} , Kt = {z ∈ H : t ≥ ζ (z)} .
4.2 Inversion of the Loewner Equation
We have just seen that any LGP family will induce a continuous function ξ on [0,∞) throughthe so-called Loewner transform. Now, we consider the inverse problem, where we start upwith a continuous function ξ. We then pursue to recover a family of functions t 7→ ft (z) thatsatisfies the Loewner equation, and in turn prove that z 7→ ft (z) are mapping-out functionsfor compact hulls. These hulls forms an LGP family, and will in fact have ξ as Loewnertransformation. It turns out that for each ξ, a unique such family exist.
We start by proving existence of solutions to the Loewner equation, and present some prop-erties that it posses. The reader will note that we generalize quite a bit compared to whatour actual interest is. Instead of limiting ourselves to the half-plane we will be solving foressentially all z ∈ C, and also treat ξ functions which are defined on the whole real line.
Theorem 4.5.Let ξ : R→ R be a continuous function. The following holds:
1. For all z 6= ξ (0) there is a solution t 7→ ft (z) to the differential equation
∂tft (z) =2
ft (z)− ξ (t), f0 (z) = z. (14)
on a maximal interval t ∈(β (z) , ζ̃ (z)
)containing 0. The solution is unique there,
and it holds that |ft (z)− ξ (t)| → 0 for t→ ζ̃ (z) whenever ζ̃ (z)
The Loewner Transform 32
Proof.Define F (t, w) = 2
w−ξ(t) as a map on D := {(t, w) ∈ R× C : w 6= ξ (t)}. Pick any z 6= ξ (0).Set (t0, w0) = (0, z). By A. 23 there is a unique solution ft (z) on a maximal interval, here
denoted as(β (z) , ζ̃ (z)
). Assume now that ζ̃ (z) 0 on[0, ζ̃ (z)
), so integrating gives
|ft (z)| ≤∫ t
0
|∂sfs (z)| ds < Ct < Cζ̃ (z) ,
which is a contradiction. On the other hand, for fixed z ∈ C\ {ξ (0)} we define KN ={(t, w) ∈
[0, ζ̃ (z)
]× C : 1/N ≤ |w − ξ (t)| ≤ N
}. The set is closed since it is the primage
of a closed set through a continuous function. It is also bounded because ξ([
0, ζ̃ (z)])
is abounded set, and therefore KN is compact. A. 23 ensures that (t, ft (z)) leaves KN for goodat some point LN ∈
[0, ζ̃ (z)
)meaning that
|ft (z)− ξ (t)| < N−1 or |ft (z)− ξ (t)| > N, ∀t ∈[LN , ζ̃ (z)
).
Trivially, if the graph leaves KN for good through the ”upper” boundary for N = 1 then itmust do so for all N ∈ N and likewise from below. If it leaves from above, it follows that|ft (z)− ξ (t)| → ∞ for t → ζ̃ (z) ∀N ∈ N. Since ξ
([0, ζ̃ (z)
])is bounded, it follows that
|ft (z)| → ∞ as well, in contradiction with what was established in the previous paragraph.It follows that |ft (z)− ξ (t)| leaves KN for good from below for all N ∈ N, and therefore itmust go to 0 for t→ ζ̃ (z).
The second statement is immediate from A. 23. To prove the third statement, note that theimaginary part of the Loewner equation is
∂vs = −2vs
(us − ξ (t))2 + v2s, v0 = =z.
Note that ∂svs 6= 0 if and only if vs 6= 0, and in that case they have opposite signs. For z ∈ Hwe have v0 > 0. Simple analysis now gives that vs is strictly decreasing on some interval(β (z) , t0), for t0 ≤ ζ̃ (z) and then constant equal to 0 on
[t0, ζ̃ (z)
). If we falsely assume
t0 < ζ̃ (z), then the real part of the Loewner equation reduces to ∂sus = 2 (us − ξ (s))−1 fors ∈
[t0, ζ̃ (z)
). However, the differential equation in D∗ = {(t, x) ∈ R2 : ξ (t) 6= x} given by
∂tU (t) =2
U (t)− ξ (t), U (t0) = u (t0)
has a unique solution U on some interval (t0 − ε, t0 + ε). On that interval U (t) + 0i is asolution to the Loewner equation, whose graph crosses the graph of ut + ivt at t = t0, butdiffers at (t0 − ε, t0). This contradicts A. 23.
The Loewner Transform 33
The same argument gives the result for z ∈ −H. Note that this in particular implies that|ξ (t)− ft (z)| cannot converge to 0 for t → β (z), and thus an argument analogous to theone in the previous paragraph shows that β (z) = −∞.
For the fourth statement, it is immediate from A. 23 that z 7→ gt (z) is injective for allt ∈ R. It is also clear that H ′t = H for t ≤ 0. It remains to prove that ft maps H ′t onto Hfor t ≥ 0. For such t and w ∈ H there is a solution l : (a, b)→ C to the equation
∂sl (s) =2
l (s)− ξ (s)l (t) = w, (15)
with graph in D and defined on some maximal interval (a, b) containing t. As before, weobtain that s 7→ =l (s) is strictly decreasing, a = −∞ and l ((−∞, t]) ⊂ H. Since b > t ≥ 0we have 0 ∈ (a, b), so our solution to (14) with z = l (0) coincides with l by uniqueness. Thisimplies that t < ζ̃ (l (0)) and l (0) satisfies ft (l (0)) = l (t) = w.
The continuous function ξ is usually called the driving function, and the solution (ft) iscalled the Loewner flow. It is immediate that ft only depends of ξ (s) for s between 0 and t,meaning that all results just proven still hold with obvious modifications if we restrict ξ to a[0,∞)→ R map. We now connect ft, H ′t with the objects defined in the previous chapters.Put K ′t = H\H ′t.
Theorem 4.6.Let ξ : [0,∞) → R be a continuous function. With notation from the previous theorem, thefollowing is true
1. The induced family of sets (K ′t)t≥0 := H\H ′t is a LGP family , parametrized by capacity.
2. Let gt be the mapping-out function for K′t. It holds that ft = gt.
3. It holds that ξ is the Loewner transform for the family (K ′t)t≥0.
◦
Proof.We start by introducing some notation. Define the map α∗t (z) = ft (z)−z, fix T ≥ 0 and de-fine r = supt≤T |ξ (t)− ξ (0)|∨
√T . Our first goal is to estimate α∗t (z) and zα
∗t (z). Take R ≥
4r and z ∈ H such that |z − ξ (0)| ≥ R. Define the time t′ := inf{t < ζ̃ (z) : |α∗t (z)| = r
}∧T ,
where t′ = ζ̃ (z) in case the set is empty. Indeed we note that t 7→ |α∗t | is continuous andmaps 0 to 0, therefore |αt (z)| ≤ r for t < t′ and thus the reverse triangle inequality gives
|ft (z)− ξ (t)| = |ft (z)− z + z − ξ (0) + ξ (0)− ξ (t)|≥ |α∗t (z)| − |z − ξ (0)| − |ξ (0)− ξ (t)|≥ R− 2r.
The Loewner Transform 34
Aside from the estimate itself, this also gives that t′ < ζ̃ (z) since |ft (z)− ξ (t)| 6→ 0 fort→ t′. Note that the Loewner equation gives the following estimate for t ≤ t′
|α∗t (z)| ≤∫ t
0
∣∣∣∣ 2fs (z)− ξ (s)∣∣∣∣ ds ≤ t 2R− 2r ≤ t′r . (16)
Analogously, using the dull t =∫ t
0fs(z)−ξ(s)fs(z)−ξ(s)ds, we get
|zα∗t (z)− 2t| ≤ 2∫ t
0
∣∣∣∣z − fs (z) + ξ (s)fs (z)− ξ (s)∣∣∣∣ ds. (17)
Since s ≤ t′ we obtain an estimate for the integrand
|z − fs (z) + ξ (s)| ≤ |α∗t (z)|+ |ξ (s)− ξ (0)|+ |ξ (0)| ≤ 2r + |ξ (0)|
which we insert into (17) to get
|zα∗t (z)− 2t| ≤t |4r + 2ξ (0)|
R− 2r(18)
for t ≤ t′. These are the desired estimates for α∗, and we may move on to show that K isa compact hull i.e. that it is bounded. First we note that it cannot be true that t′ < Tbecause the first estimate and the definition of r would imply
r = |α∗t′ (z)| ≤t′
r<T
r≤ r
2
r= r.
So t′ = T and T < ζ̃ (z) as well. This proves that z ∈ HT whenever |z − ξ (0)| ≥ R. We nowset R = 4r and conclude
z ∈ K ′T ⇒ |z − ξ (0)| < 4r. (19)
In particularKT is bounded by 8r. Moreover, choosingR large in (18) gives that |zα∗t (z)− 2t|becomes small for any sufficiently large z, and hence α∗t (z)
z→∞→ 2t for all t ≤ T . Since Twas chosen arbitrary, it holds for all t ≥ 0. In accordance with definition 2.3, we may denoteft as the mapping-out function of K
′t and conclude that aKt = 2t.
We now prove that our family of hulls has the LGP property, and it will turn out that the de-sired Loewner transform pops out for free. Fix t ≥ 0, and recall that K ′t,t+h = gt
(K ′t+h\K ′t
)is a compact hull. It is directly verified that the driving function ξ̃ : h 7→ ξ (t+ h) generatesthe Loewner flow (gt,t+h)h≥0, and the family of hulls
(K ′t,t+h
)h≥0. Now we may apply (19)
for z ∈ K ′t,t+h to get
|z − ξ (t)| =∣∣∣z − ξ̃ (0)∣∣∣ < 4r̃ = 4 sup
s∈[0,h]
∣∣∣ξ̃ (s)− ξ̃ (0)∣∣∣ ∨√h = 4 sups∈[0,h]
|ξ (t+ s)− ξ (t)| ∨√h.
This leads to
Rt,t+h < 8 sups∈[0,h]
|ξ (t+ s)− ξ (t)| ∨√h.
Since ξ is uniformly continuous on compacts of [0,∞), the LGP property follows. Moreover,it follows that any sequence zn ∈ Kt,t+ 1
nconverges to ξ (t), proving that ξ (t) ∈ Kt,t+h for all
h ≥ 0, which per definition makes ξ the Loewner transform of (K ′t)t≥0.
The Loewner Transform 35
And hence we have achieved
Proposition 4.7.Let K : C [0,∞)→ K be the map that takes ξ to the LGP family (K ′t) as given in theorem 4.6.It holds that K is a bijection, with inverse given as the Loewner transform from theorem 4.3.
◦
Since we already saw that ζ (z) = ζ̃ (z) for z ∈ H, we will drop the tilde-notationcompletely, and write ζ (z) for all z ∈ C\ {ξ (0)}. We have concluded that there is no longerreason for distinguishing between f and g on H ; they are equal. However, the Loewnerequation may be solved on a larger set Ct and gt extends by reflection to a larger set H
∗t . We
therefore ask how the reflected domain H∗t is connected to the larger set where the Loewnerequation has a solution. It turns out that Ct is preciesly equal to H
∗t . The following corollary
is crucial for the theory to come.
Corollary 4.8.Let ξ : [0,∞)→ R be a driving function. Let gt := H∗t → C be the extension by reflection ofthe mapping-out functions induced by ξ through theorem 4.6.
1. It holds that Ct = H∗t for all t ≥ 0. In particular the bi-implication ζ (x) ≤ t⇔ x ∈ Kt
is true for all x ∈ H\ {ξ (0)} , t ≥ 0.
2. The family of hull closures(Kt)t≥0 is (almost) right-continuous as well:⋂
s>t
Ks = Kt ∀t > 0,⋂s>0
Ks = {ξ (0)} .
3. For ξ (0) < x < y it holds that ζ (x) ≤ ζ (y). For y < x < ξ (0) the reverse inequalityholds.
◦
Proof.We have already seen that H ′t = Ct ∩ H = Ht. For any z s.t. z ∈ H and t < ζ (z) we maywrite gt (z) = gt (z) = ut (z)− ivt (z) which is clearly differentiable with respect to t and
∂tgt (z) = ∂tgt (z) =2
gt (z)− ξ (t)=
2
gt (z)− ξ (t), ∀t < ζ (z) .
It is immediate that ζ (z) is maximal so ζ (z) = ζ (z) . It follows that z ∈ Ct ⇔ z ∈ Ct forall z ∈ H meaning that H∗t and Ct must agree on non-reals. It is clear that Ct ∩ R ⊂ H0t soonly the reverse inclusion remains.Fix t > 0 and x ∈ H0t . We are to prove that ζ (x) > t. This is obvious if ζ (x) = ∞, sowe may assume that ζ (x) is finite. Thus it will suffice to prove that s 7→ gs (x) has not yettouched ξ (s) up until time t. First we claim that gs (x) 66= ξ (s) for any s < t. Recall thatfor any s > 0, ξ (s) is the unique point in
⋂h>0Ks,s+h. Let rn ↓ s strictly, with r1 ∈ (s, t)
and take zn ∈ Krn\Ks. If our claim is false then gs (zn) → ξ (s) = gs (x) for n → ∞. Since
The Loewner Transform 36
gs is homeomorphic on H0s ⊃ H0t we have zn → x. On the other hand we have (zn) ⊂ Kt
and hence
x = limnzn ∈ Kt
which is a contradiction. Therefore gs (x) 6= ξ (s) for s < t. It remains to prove thatgt (x) 6= ξ (t). Indeed, recall the continuity estimate
|gt (z)− gs (z)| ≤ 3Rs,t, s < t, z ∈ H∗t . (20)
Take an interval (b−, b+) containing x, whose closure lies in H0t and define c± := gt (b±).
Recall that mapping-out functions are univalent on H∗t and therefore it must be strictlymonotone on [b−, b+]. Assume without loss of generalization that c+ > c−. Notice that (20)along with the LGP implies that
∀ε > 0∃s0 > 0∀s > s0 : gs (b+) > c+ − ε, gs (b−) < c− + ε.
Take ε < min (|gt (x)− c+| , |gt (x)− c−|) and choose s0 > 0 in accordance with the statementabove. The maps r 7→ gs (r) must be strictly increasing on [b−, b+] as well since gs (b+) >gt (x) > gs (b−) for all s ∈ (s0, t]. It follows that
gt (x) ∈ (c− + ε, c+ − ε) ⊂⋂
s∈(s0,t]
gs ([b−, b+]) .
cc +- c+c--ε -εgt(x)
Figure 3: An illustration of the argument. The right and left blue line segment is the possiblevalues for gs (b±), respectively.
Hence, if the claim was false then there would be an R-neighbourhood of ξ (t) containedin the image of all gs maps s ∈ (s0, t] . Since ξ is continuous it would imply existence ofy ∈ H0t , s > s0 such that gs (y) = ξ (s) . The previous argument now gives a contradiction.In total we have concluded that gs (x) 6= ξ (s) for s ∈ [0, t] proving that ζ (x) > t.
For the second statement, take z ∈ H\ {ξ (0)} and s > t > 0. The previous result givesthat z ∈ Ks\Kt if and only if t < ζ (z) ≤ s. Letting s ↓ t gives the result
For the third statement, assume without loss of generalization that y > x > ξ (0). Since gtis univalent for all t < ζ (x) ∧ ζ (y) we must have that gt (y) − gt (x) 6= 0 for such t. Butt 7→ gt (y)− gt (x) is continuous with a positive value at t = 0 so
(gt (y)− ξ (t))− (gt (x)− ξ (t)) = gt (y)− gt (x) > 0, ∀t < ζ (x) ∧ ζ (y) . (21)
Note also that x, y > ξ (0) implies that g0 (x)−ξ (0) , g0 (y)−ξ (0) > 0, so the same argumentgives that gt (x)−ξ (t) , gt (y)−ξ (t) > 0. That, combined with (21) gives that |gt (y)− ξ (t)| >|gt (x)− ξ (t)| for all t < ζ (x) ∧ ζ (y), leading to ζ (x) ≤ ζ (y).
The Loewner Transform 37
Note these properties are exclusively for LGP families. Consider for example the familyof hulls (Kt) given by
Kt =
{(it] t ≤ 1(2 + i (t− 1)] t > 1
If it were an LGP family, the Loewner transform would satisfy ξ0 = 0. Since 2 ∈ K2, 1 /∈ K2it would follow from 1) that ζ (1) > 2 ≥ ζ (2) which violates 3). Moreover, the hull closuresare not right continuous ; K1 6= K1+.
The Schramm-Loewner Flow 38
5 The Schramm-Loewner Flow
We now introduce stochasticity of the Loewner flow. Consider the background space (Ω,F , P ).Let B be a two-sided Brownian motion with volatility κ, which is defined to be a stochasticprocess indexed on R satisfying that t 7→ Bt and t 7→ B−t are two independent Brownianmotions with variance parameter σ2 = κ. Define F+t = Ft as the enlargement of σ (Bs)s∈[0,t]satisfying the usual hypothesis, and define F−t analogously. Due to reflection, it will suf-fice to discuss the distribution of g restricted to (t, z) ∈ R × H. Let G be the space oftriples (β, ζ, g) such that ζ, β are lower/upper semicontinuous H → ± [0,∞] maps and g isa R × H → C map which is continuous and H-valued on {(t, z) |t ∈ (β (z) , ζ (z))} and setto a dummy-variable −i everywhere else. We have seen that the operator G, which sends acontinuous function ξ to the Loewner flow, is a well defined C (R,R)→ G injection. EquipC (R,R) with the usual σ-algebra of compact-uniform convergence, and put an appropriateσ-algebra G on G as well.Definition 5.1.The Schramm-Loewner flow is the family of laws G (Wκ)κ>0, whereWκ is the Wiener measurewith volatility κ > 0. ◦
A similar construction leads to the Schramm-Loewner distribution on compact hulls,given by the family of laws (K (Wκ))κ>0. Due to material-selection, the author has chosen toleave most questions of measurability outside the scope of this dissertation. In fact we willnot even spend time discussing an appropiate construction of G. Nevertheless, we will with-out remorse discuss probabilities such as P (gt (z) ∈ A) for A ∈ B (C), and use argumentssuch as ξ1
D= ξ2 ⇒ G (ξ1) = G (ξ2) and ξ1 |= ξ2 ⇒ G (ξ1) |= G (ξ2).
We will now see a sketch of how to prove that t 7→ gt (z) is adapted. In particular we willshow the important and intuitively clear result stating that ζ (z) is a stopping time. Duringthe proof it will also be revealed how the Schramm-Loewner flow is connected to the Besselflow.
Proposition 5.2.Let z ∈ H. The process t 7→ gt (z) 1(β(z),ζ(z)) (t) is adapted to (Ft) and ζ (z) is a stoppingtime. ◦The notation means that the process is set to 0 outside the interval [0, ζ (z)), where it isformally not defined.
sketch of proof.We show that
(gt (z) 1[0,ζ(z))
)t≥0 is adapted to (Ft)t≥0. Let z ∈ H and consider the SDE
dYt = dWt + fn (Yt) dt, Y0 = z, (22)
where Wt = −Bt/√κ and the map fn (w) is given by 2 (κw)
−1 for w ∈ C\B (0, n−1) andsome Lipchitz continuation2 on B (0, n−1). By theorem 9.1 and 9.14 [Jac08] the SDE has a
2The real and imaginary part may for example be extended by
The Schramm-Loewner Flow 39
solution t 7→ Xn,t which is strong, unique u.t.i and adapted to (Ft). Define τn to be the firsttime Xn,t hits B (0, n
−1) . Since gt (z) solves the Loewner equation, a direct calculation showsthat Xt (z) = (gt (z
√κ)−Bt) /
√κ solves (22) for t ∈ [0, τ (z) ∧ τn) where τ (z) = ζ (z
√κ).
Since t 7→ gt (z√κ) is the unique solution to the Loewner equation and τ (z) is a maximal, it
follows that τn < τ (z) and Xn,tu.t.i= Xt on [0, τn]. Define the stopping times τ̃ := supn∈N τn,
τ0 = 0 . It is easily seen that
X̃t =∞∑k=1
Xk,t1[τk−1,τk) (t)
solves the SDE on the interval [0, τ̃), and that X agrees with X̃ there u.t.i. Since X̃t is a seriesof (Ft)-adapted processes, it is itself an adapted process. Note that τ (z) is characterized byXt exists and is positive for t < τ (z), but converge to 0 for t→ τ (z). Since |Xτn| = 1/n→ 0for n→∞ it follows that τ̃ = τ (z), proving that τ (z) is a stopping time. Finally, note thatgt (z√κ) 1[0,τ(z)) (t) = (Xt
√κ+Bt) 1[0,τ(z)) (t) which concludes the proof.
We shall return to the Bessel SDE at a later point. It is intuitively clear that the Loewnerflow with a Brownian motion as driving function will inherit many of its properties. Thefollowing proposition shows that this is the case. In the following, set Ht = H, Kt = Ø fort < 0.
Proposition 5.3.Let (gt)t∈R be the Schramm-Loewner flow corresponding to the two-sided Brownian motionB.
1. The flow satisfies a scale-invariance property; the process g̃ : (t, z) 7→ c−1/2gct (√cz)
has the same distribution as (t, z) 7→ gt (z), and the family of scaled hulls c−1/2Kct hasthe same distribution as (Kt)
2. The flow has the domain Markov property; for any t0 ≥ 0 the family of translatedremainder hulls given by K
(t0)t = Kt0,t0+t−Bt0 for t ≥ 0 and Ø for t ≤ 0 has the same
distribution as (Kt) and
g(t0) : (t, z) 7→ gt0,t0+t (z +Bt0)−Bt0
has the same distribution as (t, z) 7→ gt (z). Moreover, when restricting the flow tot ≥ 0, it is independent of Ft0, and t0 may be replaced by a F+t stopping time.
3. The flow is invariant under reflection through the imaginary axis; The family of re-flected hulls K̂t = {x+ iy ∈ H : −x+ iy ∈ Kt} has the same distribution as (Kt) and
ĝ : (t, z) 7→ −u (−x+ iy) + iv (x+ iy)
has the same distribution as (t, z) 7→ gt (z).
◦
The Schramm-Loewner Flow 40
Proof.By self-similarity, t 7→ Bt has the same distribution as t 7→ c−1/2Bct. It then holds that
G (B)D= G
(t 7→ c−1/2Bct
)Note that t 7→ g̃t (z) is well-defined and differentiable exactly when ct < ζ
(c1/2z
), corre-
sponding to z ∈ c1/2Kct. In that case we have
∂tg̃t (z) = c1/2∂ctgct
(c1/2z
)=
2
c−1/2gct (c1/2z)− c−1/2Bct
=2
g̃t (z)− c−1/2Bct, g̃0 (z) = z.
This shows the first point.
For the second point, we use that
G (B)D= G (t 7→ Bt+t0 −Bt0) .
We verify that G (t 7→ Bt+t0 −Bt0) = g(t0) by insertion into the Loewner equation. Note thatg(t0) is differentiable exactly when t0+t < ζ
(g−1t0 (z +Bt0)
)corresponding to z ∈ Kt,t0+t−Bt0
for t ≥ 0 and K(t0)t = Ø for t ≤ 0. In that case we have
∂t (gt0,t0+t (z +Bt0)−Bt0) = ∂t(gt0+t
(g−1t0 (z +Bt0)
)−Bt0
)=
2
gt0+t(g−1t0 (z +Bt0)
)−Bt0+t
=2
gt0+t(g−1t0 (z +Bt0)
)−Bt0 − (Bt0+t −Bt0)
=2
g(t0)t (z)− (Bt0+t −Bt0)
, g(t0)0 (z) = z.
This proves the first claim. If we restrict our attention to the t ≥ 0 process, we may gothrough the exact same argument, this time considering t0 as a stopping time and using thestrong Markov property.
For the last statement, we use that
G (B)D= G (t 7→ −Bt) .
By insertion it is verified that (t, z) 7→ ĝt (z) is differentiable exactly when t < ζ (−x+ iy),and in that case t 7→ ĝt (z) solves the real, and imaginary part of the Loewner equation withdriving function t 7→ −Bt, and with correct boundary condition.
Existence of the Schramm-Loewner Evolution 41
6 Existence of the Schramm-Loewner Evolution
The goal of this section will be to derive the following theorem.
Theorem 6.1 ( Existence of The SLE Trace).Let (gt) be the Schramm-Loewner flow induced by B ∼ BM (0, κ). Almost surely the followingis true: The limit
γt := limz→0
g−1t (Bt + z)
exists for all t ≥ 0, and forms a continuous curve in H. Furthermore, we may recover Ht asthe unbounded component of H\γ [0, t]. ◦
Definition 6.2.The Schramm-Loewner Evolution (SLE) , or the Schramm-Loewner trace with parameter κ,is the continuous stochastic process γ from theorem 6.1. ◦
Unfortunately, and a bit curiously , there has yet to be found a proof of this result,that treats all κ > 0 simultaneously. The problem appears at κ = 8, which is treated asa complicated exception in the literature.We shall only prove the result for κ 6= 8, and wewill follow the original proof found at [SR05]. In order to prove existence of the SLE, wefirst prove theorem 6.4, which gives a sufficient condition for when an inverse Loewner flowextends continuously to R. Then we prove theorem 6.5 which ensures that the Loewner flowinduced by a Brownian motion, almost surely satisfies this condition. The latter argumentis long, so we will split it into several lemmas. We immediately see that if theorem 6.1 holds,then the γ process inherits the invariance properties from proposition 5.3
Proposition 6.3.Let (γt)t≥0 be an SLEκ.
1. The scaled process c−1/2γct is also an SLEκ process.
2. For any finite stopping time τ with respect to (Ft), the process(γ
(τ)t
)t≥0 :=
(g−1τ,τ+t (Bτ+t)−Bτ
)t≥0
is another SLEκ, independent of Fτ . Moreover, this transform allows inversion
γτ+t = g−1τ
(γ
(τ)t +Bτ
).
3. The reflection of (γt)t≥0 through the imaginary axis is an SLEκ.
◦
Proof.We will not prove this, but the invariance results should be intuitively evident, when recallingthe invariance results for the flow. Regarding the inversion in the second part, note that g−1τ
Existence of the Schramm-Loewner Evolution 42
and the collection g−1τ,τ+t t ≥ 0 extends continuously to z ∈ H with probability 1. This leadsto
g−1τ
(γ
(τ)t +Bτ
)= g−1τ
(g−1τ,τ+t (Bτ+t)
)= lim
z→Bτ+tg−1τ
(g−1τ,τ+t (z)
)= lim
z→Bτ+tg−1τ+t (z)
= γτ+t.
6.1 Proof of the SLE Existence
Define the map f̂ : (t, z) 7→ g−1t (z + ξ (t)) for t ≥ 0, z ∈ H. For fixed t this is a univalentH → H map. As mentioned above, the following theorem is non-stochastic and applies toLoewner flows induced by an arbitrary driving function.
Theorem 6.4.Let ξ be a driving function. Assume that the vertical limit
γ (t) := limy→0
f̂t (iy)
exists for all t ≥ 0 and forms a continuous map. Then g−1t extends continuously to H,∂Ht ∩ H ⊂ γ [0, t] and Ht is the unbounded connected component of H\γ ([0, t]) for everyt ≥ 0. ◦
Proof.For t > 0 we define the set S (t) as all limitpoints of sequences g−1t (wn), where (wn) ⊂ H isa sequence satisfying wn → ξ (t). First we prove that
S (t) ⊂ γ [0, t] ∀t ≥ 0. (23)
Take z0 ∈ S (t0) for t0 ≥ 0. We may then find a sequence wn → ξ (t0) such that zn :=g−1t0 (wn)→ z0. Recall from A. 5 that z0 ∈ ∂Ht0 . if we falsely assume that z0 ∈ H0t0 then gt0is homeomorphic at z0 so ξ (t0) = limn→∞ gt0 (zn) = gt0 (z0). Corollary 4.8 and theorem 4.5would give the false inequality
0 = |ξ (t0)− ξ (t0)| = |gt0 (z0)− ξ (t0)| > 0.
Therefore z0 ∈ Kt0 corresponding to B (z0, �) ∩Kt0 6= Ø for all � > 0. Fix some � > 0 anddefine the set of timepoints
A ={t ≥ 0 : Kt ∩B (z0, �) = Ø
}.
Indeed, 0 ∈ A and A is bounded by t0, so we may define t′ = supA. Define p as zn forsome n such that zn ∈ Ht0 ∩ B (z0, �). We claim that Kt′ ∩ B (z0, �) is not empty. Indeed,
Existence of the Schramm-Loewner Evolution 43
per construction of t′, it must hold that Ks ∩ B (z0, �) 6= Ø for all s > t′. Take a sequencepn ∈ Ksn ∩ B (z0, �) for some sn ↓ t. The sequence lies in the compact set Ks1 so we mayassume that it has a limit p′. For any fixed m ∈ N (pn) lies in the closed set Ksm ∩B (z0, �)eventually, so p′ ∈ Ksm as well, and since m was chosen arbitrarily we get that
p′ ∈⋂m∈N
B (z0, �) ∩Ksm .
As we saw in corollary 4.8, the closed hulls are right continuous as well, so p′ ∈ Kt′∩B (z0, �).Consider the line segment [p, p′] and define p′′ as the first point in Kt′ .
z0
p
p
p''
'
Figure 4: An illustration of some of the introduced points. The purple area is Kt′ and thesalmon-colored area corresponds to Kt0 .
We apply A. 8 to achieve that gt′ ([p, p′′)) is a curve in H with endpoint in some x (t) ∈ R.
Since p′′ ∈ Kt′ it holds that ζ (p′′) ≤ t′ by corollary 4.8. Strict inequality cannot hold, becausewe would have some t close to t′ so that ζ (p′′) ≤ t, implying p′′ ∈ Kt. But the point p′′,along with the whole line-segment [p, p′] lies in the convex set B (z0, �) so p
′′ ∈ Kt∩B (z0, �),which contradicts the definition of t′. We conclude that ζ (p′′) = t′, and we can proceed toa new claim, namely
x (t) = ξ (t′) = ξ (ζ (p′′)) .
Indeed, for any z ∈ [p, p′′) , t < ζ (p′′) we have
|x (ζ (p′′))− ξ (ζ (p′′))| ≤∣∣x (ζ (p′′))− gζ(p′′) (z)∣∣+ ∣∣gζ(p′′) (z)− gt (z)∣∣+ |gt (z)− gt (p′′)|
(24)
+ |gt (p′′)− ξ (t)|+ |ξ (t)− ξ (ζ (p′′))| .
Existence of the Schramm-Loewner Evolution 44
Fix N ∈ N. The continuity estimate gives∣∣gζ(p′′) (z)− gt (z)∣∣ < 3Rt,ζ(p′′) ∀z ∈ H∗ζ(p′′).for all z ∈ Ht. This allows us to choose δ2 so that the second term is smaller than 15N fort ∈ (ζ (p′′)− δ2, ζ (p′′)). By continuity of ξ at t = ζ (p′′) a
