Defining SLE in multiply connected domains with the Brownian loop measure
aa r X i v : . [ m a t h . P R ] A ug DEFINING SLE IN MULTIPLY CONNECTED DOMAINS WITHTHE BROWNIAN LOOP MEASURE
GREGORY F. LAWLER
Abstract.
We define the Schramm-Loewner evolution (
SLE κ ) in multiplyconnected domains for κ ≤ Introduction
The Schramm-Loewner evolution (
SLE ) is a conformally invariant or confor-mally covariant family of measures on curves in the plane. It was proposed bySchramm [19] as a candidate for the scaling limit of loop-erased walk and per-colation interfaces, and it has turned out to be the crucial tool in the rigorousdevelopment of two-dimensional critical phenomenon. Before
SLE , there had beenmuch theoretical, but mathematically nonrigorous, development using conformalfield theory.In conformal field theory, the standard parameter to characterize a field is thecentral charge c . There is a major difference between c ≤ c >
1, and
SLE appears in the former case which is all we consider in this paper. The parameterfor
SLE is denoted κ >
0. For each c <
1, there are two values of κ , one less thanfour and one greater than four, given by c = (6 − κ )(3 κ − κ . The smaller value corresponds to the simple curve case, and we concentrate on thisin this paper. For c = 1, κ = 4 is a double root which also corresponds to simplecurves. Important examples are κ = 2 , c = − κ = 8 / , c = 0(self-avoiding walks), κ = 3 , c = 1 / κ = 4 , c = 1(interfaces of free fields). In all cases, but for self-avoiding walk, SLE has beenproved to be the scaling limits of the models [16, 23, 21] ♣ The letter c is standard in the physics literature for central charge. Since we use c forarbitrary constants, it is not a good choice for a parameter. Our compromise is to use abold-face c . Another conformally invariant measure on (in this case, nonsimple) curves in theplane is given by Brownian motion. A variant of this measure, called the Brownianloop measure, arises in the study of
SLE [12, 18]. This is a σ -finite measure onnonsimple curves that arises as a scaling limit of a random walk loop measure, see[17] and [15, Chapter 9]. It is closely related to the determinant of the Laplacian Research supported by National Science Foundation grant DMS-0907143. and the Gaussian free field, see, e.g., [4], but we will only need to view it as ameasure on paths. The key properties of the measure are conformal invariance andthe restriction property.In this paper we view
SLE κ as a (positive) measure µ D ( z, w ) on curves (moduloincreasing reparametrization) in a domain D of total mass Ψ D ( z, w ) connectingtwo distinct points z, w . Here z, w can be interior points or boundary points butin the latter case we make some smoothness assumptions on the boundary. Weexpect these curves to arise as normalized limits of measures on lattice curves. If0 < Ψ D ( z, w ) < ∞ , we can normalize the measure to produce a probability measurethat we denote by µ D ( z, w ). There are various assumptions we can make on themeasures. We will be more precise later, but let us discuss them now. The first isconformal covariance: • Conformal covariance.
There exist boundary and interior scaling expo-nents b, ˜ b such that if f : D → f ( D ) is a conformal transformation, f ◦ µ D ( z, w ) = | f ′ ( z ) | b z | f ′ ( w ) | b w µ f ( D ) ( f ( z ) , f ( w )) , where b ζ = b if ζ ∈ ∂D and b ζ = ˜ b if z ∈ D .This implies conformal invariance of the probability measures, f ◦ µ D ( z, w ) = µ f ( D ) ( f ( z ) , f ( w )) . If one is only considering the probability measures, then one does not need to makesmoothness assumptions at the boundary. The domain Markov property below usesthe probability measures for nonsmooth boundary points.There are three other assumptions we will discuss. It turns out that they areredundant, so we do not need to make all of them assumptions, but this is notobvious. • Reversibility.
The measure µ D ( w, z ) can be obtained from µ D ( z, w ) byreversing the paths. • Domain Markov property . In the probability measure µ D ( z, w ) , givenan initial segment of the curve γ t = γ (0 , t ], the conditional distribution ofthe remainder of the curve is µ D \ γ t ( γ ( t ) , w ) . • Boundary perturbation.
Suppose D ⊂ D and the domains agree inneighborhods of z, w . Then µ D ( z, w ) is absolutely continuous with respectto µ D ( z, w ). In fact, if γ is a curve connecting z and w in D , then theRadon-Nikodym derivative is given byexp n c m D ( γ, D \ D ) o , where m D ( γ, D \ D ) denotes the (Brownian) loop measure of loops in D that intersect both γ and D .Schramm [19] studied the probability measures µ D ( z, w ) where z ∈ ∂D and w ∈ D or w ∈ ∂D . He showed that for simply connected D , there is only aone-parameter family of measures satisfying conformal invariance and the domainMarkov property. He used κ as the parameter and these are now called radial andchordal SLE κ (in D from z to w ), respectively. It is known [20, 2] that for κ ≤
4, themeasure is supported on simple curves of Hausdorff dimension d = 1 + κ ∈ (1 , ].The following has been proved for SLE κ , < κ ≤ LE IN MULTIPLY CONNECTED DOMAINS 3 • Let b = 6 − κ κ , ˜ b = b ( κ − b + c . Let Ψ H (0 ,
1) = 1 , Ψ D (1 ,
0) = and define Ψ D ( z, w ) for other simply con-nected domains by the scaling ruleΨ D ( z, w ) = | f ′ ( z ) | b | f ′ ( w ) | b w Ψ f ( D ) ( f ( w ) , f ( w )) , where b w = b if w ∈ ∂D and b w = ˜ b if w ∈ D . Then [12, 18, 9] if µ D ( z, w ) = Ψ D ( z, w ) µ D ( z, w ), the family { µ D ( z, w ) } restricted to simplyconnected domains satisfies conformal covariance, domain Markov property,and the boundary perturbation rule. • If w ∈ ∂D , then [24] µ D ( w, z ) is the same as the reversal of µ D ( z, w ).In the chordal case, Ψ D ( z, w ) = H ∂D ( z, w ) b where H ∂D ( z, w ) denotes a multipleof the boundary Poisson kernel. This follows from the scaling rule for the kernel, H ∂D ( z, w ) = | f ′ ( z ) | | f ′ ( w ) | H ∂f ( D ) ( f ( z ) , f ( w )) . If w ∈ D , the Poisson kernel satisfies H D ( w, z ) = | f ′ ( z ) | H f ( D ) ( f ( w ) , f ( z )) , and hence Ψ D ( w, z ) is not given by a power of the Poisson kernel. If κ = 2, for which b = 1 , ˜ b = 0, the partition function is given by the boundary Poisson kernel (chordalcase) or Poisson kernel (radial case). One can also see this from the relationshipwith loop-erased random walk.In his argument, Schramm uses the fact that if one slits a simply connected do-main D at its boundary then the resulting domain D \ γ t is also simply connectedand hence by the Riemann mapping theorem is conformally equivalent to the orig-inal domain. If D is not simply connected, or D is “slit on the inside”, this is nolonger true. For this reason, conformal invariance of the probability measures andthe domain Markov property are not sufficient to determine the measures µ D ( z, w )for nonsimply connected domains. In [14] it was suggested to use the boundaryperturbation rule to extend the definition. We continue this approach in this pa-per. There have been other approaches, see, e.g., [ ? , ? , ? , ? ], but none have directlyused the boundary perturbation rule.We will show the following. (If z or w are boundary points, we implicitly assumesufficient smoothness at the boundary.) • There is a unique way (up to some arbitrary multiplicative constants) toextend the measures µ D ( z, w ) so that it satisfies conformal covariance andthe boundary perturbation rule. • If κ ≤ / c ≤ D ( z, w ) < ∞ , and the probability measuressatisfy the domain Markov property. • If 8 / < κ ≤
4, and D is 1-connected, Ψ D ( z, w ) < ∞ .The key observation is that the restriction property for the Brownian loop measureholds for multiply connected domains. We conjecture that Ψ D ( z, w ) < ∞ for all κ ≤
4, but have not shown this. However, we prove a weaker fact. • If κ ≤ D is a simply connected subdomain and µ D ( z, w ; D ) denotesthe measure µ D ( z, w ) restricted to curves staying in D , then k µ D ( z, w ; D ) k < ∞ . • The probability measures µ D ( z, w ; D ) satisfy the domain Markov property. GREGORY F. LAWLER • If Ψ D ( z, w ) < ∞ for all k -connected domains, then the measures µ D ( z, w ),restricted to k -connected domains, satisfy the domain Markov property.The next property will follow from the definition and Zhan’s result for simplyconnected domains [24]. • The measure µ D ( w, z ) is the reversal of µ D ( z, w ).Zhan [25] recently took a different approach to extending SLE κ in the case ofan annulus. Roughly speaking, he shows that there is a unique way of defining µ D ( z, w ) for conformal annuli so that it satisfies the domain Markov property andreversibility. (Note that the combination of the two properties allows one to describeconditional distributions given both an initial segment and a terminal segment ofthe path.)In this paper, we consider our process for 1-connected domains and show that it isthe same as that defined by Zhan. In particular, reversibility of the process follows.We use the boundary perturbation rule to give an equation for the partition functionand give a somewhat more direct proof of existence of the solution. Although thispaper does not directly use the results in [25], it does use an idea from that paper.In particular, the annulus Loewner equation is used to find PDEs and the Feynman-Kac formula is used to analyze PDEs that arise.We now summarize the contents of the paper. We describe in Section 2 a modelintroduced in [8] called the λ -SAW. It is a two-parameter family of lattice models forwhich it is conjectured that there is a one-parameter subfamily of critical models.One of the parameters in [8] was denoted λ but we have chosen to set λ = − c / c = −
2) and self-avoiding walk( c = 0). This model was created after studying SLE . While we cannot provethat this has a limit at the moment (except for c = − c = 1 for which we can use current results), it is useful for heuristicunderstanding of our definition of SLE in multiply connected domains.Section 3 contains many results that are needed in the paper most of which havebeen proved elsewhere. This can be skimmed at first reading and referred backto as needed. Section 3.1 reviews facts about the Poisson kernel and sets somenotation; this is followed by discussion of the annulus version. The annulus Poissonkernel is often written in terms of theta functions. We choose instead to write thefunctions in terms of infinite sums which arise naturally when raising the annulusto the covering space of an infinite strip. The next three subsections review theimportant tools in this area:
SLE in H , the Brownian bubble measure, and theBrownian loop measure. Section 3.6 reviews the methods to analyze SLE in simplyconnected domains in terms of the Brownian loop measure and extends this idea toshrinking domains. This will allow us to view radial
SLE or annulus
SLE in termsof chordal
SLE in H where the domain is shrinking by all the translates of the path.In the case of annulus SLE we get a process that we call locally chordal
SLE κ . Wewrite this using an annulus parametrization and this leads to the annulus Loewnerequation which we write as an equation in the covering infinite strip.The definiton of SLE is given in Section 4. In the boundary to boundary case,this is essentially the same definition as in [14]. We extend this to boundary/bulkand bulk/bulk cases. One nice thing about our definition is that reversibility isimmediate, given reversibility for chordal
SLE in simply connected domains. Thereare some subtleties in defining the bulk/bulk measure in subdomains of C in termsof the measure on C , see Proposition 4.8. The definitions make use of facts about LE IN MULTIPLY CONNECTED DOMAINS 5 annulus
SLE that are discussed in the next section. The extension of the definitionto multiple paths with disjoint endpoints is immediate as in [8].The next two sections discuss the results about annulus
SLE κ . Most of theresults in this section were proved in [25], but there are some differences in ourapproach. We focus on the “crossing” case although the “chordal” case can bedone similarly as we point out. In Section 5 we study annulus SLE κ with a givenwinding number. By taking its premage under the logarithm, we can consider itas a measure on curves connecting points of an infinite strip, and we in turn cancompare this measure to chordal SLE κ in the strip. This requires comparing theloop measures in the strip to the preimage of the loop measure in the annulus.(Although the loop measure is conformally invariant, the logarithm is a multi-valued function, so some care is needed.) At an intermediate step we consider thelocally chordal SLE κ discused in Section 3. Although this latter process is not thesame as annulus SLE κ , it turns out that the partition function for annulus SLE κ can be given in terms of a functional of this process. As in [25], we can then use theFeynman-Kac theorem to write a PDE for the partition function and this allows usto show that it gives the quantity we want.Section 7 takes a different approach and derives the differential equation for thepartition function in the annulus by comparing annulus SLE κ to radial SLE κ .Smoothness of the partition function follows from the work of the previous section,so only the Itˆo formula calculation is needed. The work here shows that the processwe get is the same as the process in [25]. Our approach gives a little more than whatis stated explicitly in [25]. The annulus partition function is of the form Ψ( r, x ),which denotes the total mass of SLE κ from 1 to e − r + ix in the annulus A r = { e − r < | z | < } . The probability measure is obtained by normalization. Multiplying thepartition function by a function of r does not change the probability measure. Herewe get not only the probability measure but the correct r dependence.I would like to thank Dapeng Zhan for useful conversations.2. The lattice model
Here we describe a lattice model for random walks called the λ -SAW [8]. Forsimplicity, we will start with the bulk/bulk version in a bounded domain D . Forconvenience, we will use the integer lattice Z = Z + i Z , but the scaling limit shouldbe independent of the lattice.A self-avoiding walk (SAW) ω = [ ω , . . . , ω n ] of length n is a finite nearestneighbor path in Z such that ω j = ω k for j < k . Let | ω | = n denote the length.A rooted (random walk) loop η = [ η , . . . , η n ] of length 2 n > η = η n . Again we write | η | = 2 n for the length. An unrooted loop is an equivalence class of loops underthe equivalence relation[ η , . . . , η n ] ∼ [ η j , η j +1 , . . . , η n , η , . . . , η j ]for each j . The rooted random walk loop measure is the measure on rooted loops,which assigns measure 4 −| η | / | η | . to each loop η with | η | >
0. This induces ameasure m RW on unrooted loops called the random walk loop measure by givingeach unrooted loop the sum of the weights of the different rooted loops that givethe unrooted loop. GREGORY F. LAWLER ♣ One may think of the unrooted loop measure as assigning measure − n to each unrootedloop η with | η | = n . However, this is not exactly correct. For example, if n = 4 and η = [ x, y, x, y, x ] , then there are only two different rooted loops that generate the unrootedloop, and hence this unrooted loop has measure − n / . Suppose D is a bounded domain in C and z, w are distinct points in D . Let β, λ be fixed constants which are the parameters of the model. For each n , let L n = n − Z ∩ D and let z n , w n be points in L n closest to z, w (if there is a tie for“closest”, we can choose arbitrarily). Define the measure ν n = ν n,D,z,w on SAWs ω in L n with endpoints z n , w n which gives ω measureexp (cid:8) − β | ω | + λ m RW ( ω, D, n ) (cid:9) , where m RW ( ω, D, n ) denotes the total m RW measure of (unrooted) loops η in L n that intersect ω . Let Z n ( D ) = Z n ( D ; β, λ ) denote the total mass of the measure.This is also called the partition function .This model has two parameters but the conjecture is that there is a one-parameterfamily of critical models. Let us write λ = − c / β = β c for the corre-sponding value of β . ♣ The value of the critical β is a lattice dependent quantity. The value λ = − c / is notlattice dependent as long as we define the random walk loop measure correctly. For a givenlattice, the rooted loop measure is defined to give measure p ( η ) / | η | to every loop η where p ( η ) is the probability that simple random walk in the lattice starting at η produces the loop η . The value c is the “central charge” but we can think of it as a free parameter. Conjecture 2.1.
For each c ≤ , there corresponds a (lattice dependent) β anda (lattice independent) scaling exponent ˜ b such the measure ν n has the followingproperties. • For each bounded D and distinct z, w in D there exists Ψ ∗ D ( z, w ) ∈ (0 , ∞ ) such that Z n ∼ n − b Ψ ∗ D ( z, w ) , n → ∞ . • There exists a limit measure on simple curves ν D ( z, w ) = lim n →∞ n b ν n . • The family of measures { ν D ( z, w ) } satisfies the conformal covariance rela-tion: if f : D → f ( D ) is a conformal transformation, f ◦ ν D ( z, w ) = | f ′ ( z ) | ˜ b | f ′ ( w ) | ˜ b ν f ( D ) ( f ( z ) , f ( w )) . There is also a boundary version of this conjecture. Suppose z is a boundarypoint of D and let us assume that ∂D is analytic near z . One can define the measure ν n as above, but there are lattice issues involved. We will not deal with them hereand just state the following rough conjecture; see [ ? ] for a more precise statementincluding lattice issues. We also assume smoothness near the appropriate boundarypoints. Conjecture 2.2.
For each c ≤ , there corresponds a (lattice dependent) β and(lattice independent) scaling exponents b, ˜ b such the measure ν n has the followingproperties. LE IN MULTIPLY CONNECTED DOMAINS 7 • For each bounded D and distinct z, w in D there exists Ψ ∗ D ( z, w ) such that Z n ∼ n − ( b z + b w ) Ψ ∗ D ( z, w ) , n → ∞ . • There exists a limit measure on simple curves ν D ( z, w ) = lim n →∞ n b z + b w ν n . • The family of measures { ν D ( z, w ) } satisfies the conformal covariance rela-tion: f ◦ ν D ( z, w ) = | f ′ ( z ) | b z | f ′ ( w ) | b w ν f ( D ) ( f ( z ) , f ( w )) . Here b ζ = b or ˜ b , respectively, if ζ is a boundary point or an interior point. The conjectures are open, but let us assume that the conjectures do hold. Let ν D ( z, w ) = ν D ( z, w )Ψ ∗ D ( z, w )be the corresponding probability measures which are conformally invariant : f ◦ ν D ( z, w ) = ν f ( D ) ( f ( z ) , f ( w )) . Schramm [19] showed that if D is simply connected and z ∈ ∂D , there is only aone-parameter family of possible limit measures for ν D ( z, w ) which are now called chordal (if w ∈ ∂D ) or radial (if w ∈ D ) Schramm-Loewner evolution with param-eter κ ( SLE κ ). Analysis of
SLE [20, 12] shows that 0 < κ ≤ b = 6 − κ κ , ˜ b = b ( κ − , c = 6˜ b − b = b (3 κ − . Suppose z, w ∈ D and D ⊂ D , and let ν n , ν n be the corresponding measures asabove and L n = D ∩ n − Z , L n = D ∩ n − Z . Then if ω is a SAW in L n connecting z n and w n , ν n ( ω ) ν n ( ω ) = exp n c m RW ( ω, D, n ) − m RW ( ω, D , n )] o . As n → ∞ , the quantity on the right has a limit [17] in terms of the Brownian loopmeasure lim n →∞ [ m RW ( ω, D, n ) − m RW ( ω, D , n )] = m D ( ω, D \ D ) , where the right-hand side denotes the Brownian loop measure [18] of loops in D that intersect both ω and D \ D . Hence the limit measures should satisfy for γ ⊂ D ,(2) dν D ( z, w ) dν D ( z, w ) ( γ ) = exp n c m D ( γ, D \ D ) o . For simply connected
D, D with z ∈ ∂D , this was established in [12, 18].Schramm’s construction of SLE makes generalizations to nonsimply connecteddomains difficult. The purpose of this paper is to show that one can use the relation(2) to define it. This requires some work. While we do not prove the conjecturesstated in this section, it is helpful to remember that the definitions we give in thispaper are those of the conjectured scaling limit of the λ -SAW with λ = − c / GREGORY F. LAWLER Preliminaries
In this paper, we assume that κ ∈ (0 ,
4] and c , b, ˜ b are as in (1). We also set a = 2 κ . Constants throughout may depend implicitly on κ .3.1. Poisson kernel and related.
We establish notation and review facts aboutthe Poisson kernel. • H denotes the open upper half plane, D the open unit disk, and if r > A r = { z ∈ D : e − r < | z |} , S r = { z ∈ H : Im( z ) < r } , D r = e − r D , C r = ∂ D r . Under this notation A r = D \ D r , ∂A r = C ∪ C r . Throughout this paperwe fix ψ ( z ) = e iz and note that ψ maps S r (many-to-one) onto A r . We write + ∞ , −∞ forthe two infinite points in ∂S r . • If D is a domain, then z is ∂D -analytic if z ∈ ∂D and there is a neighbor-hood N of z and a conformal transformation φ : N → φ ( N )with φ ( z ) = 0 and φ ( N ∩ D ) = φ ( N ) ∩ H . We say that z is D -analytic if z ∈ D or z is ∂D -analytic. • If γ is a curve, we write γ t for γ [0 , t ]. • If z, w ∈ ∂D and γ : [0 , t ] → D is a curve with γ (0) = z, γ ( t ) = w weabuse notation by writing γ ⊂ D if γ (0 , t ) ⊂ D . If t < t , we write γ t ⊂ D if γ (0 , t ] ⊂ D . • If z ∈ D and w is ∂D -analytic, let H D ( z, w ) denote the Poisson kernel (thatis, the inward normal derivative of the Green’s function at w ) normalizedso that H H ( x + iy,
0) = yx + y . It satisfies the scaling rule H D ( z, w ) = | f ′ ( w ) | H f ( D ) ( f ( z ) , f ( w )) . (When writing rules like this, it will be implicitly assumed that the quanti-ties are well defined. For example, in this case z ∈ D , w is ∂D -analytic, and f ( w ) is ∂f ( D )-analytic.) Under our normalization, the probability that acomplex Brownian motion starting at z exits D at V ⊂ ∂D is1 π Z V H D ( z, w ) | dw | . • If z, w are distinct ∂D -analytic points, we write H ∂D ( z, w ) for the boundary or excursion Poisson kernel given by H ∂D ( z, w ) = ∂ n H D ( z, w ) = H ∂D ( w, z ) , where n = n z denotes the (inward) normal derivative at z . It satisfies thescaling rule:(3) H ∂D ( z, w ) = | f ′ ( z ) | | f ′ ( w ) | H ∂f ( D ) ( f ( z ) , f ( w )) . LE IN MULTIPLY CONNECTED DOMAINS 9 • If D is simply connected, there is a complex form of the Poisson kernel H D ( z, w ) such that H D ( z, w ) = Im H D ( z, w ). This is defined up to a realtranslation, and we choose the translation so that H H ( z,
0) = − z . The function f ( z ) = H D ( z, w ) can be characterized as the unique conformaltransformation f : D → H such that f ( w + ǫ n w ) = iǫ + o (1) , ǫ ↓ . • The Poisson and boundary Poisson kernel for the strip S r can be computedusing conformal invariance,(4) H ∂S r ( z,
0) = − π r coth (cid:16) πz r (cid:17) , (5) H ∂S r (0 , x ) = π r h sinh (cid:16) πx r (cid:17)i − , (6) H ∂S r (0 , x + ir ) = π r h cosh (cid:16) πx r (cid:17)i − . • If z, w are distinct boundary points of D , D ⊂ D with dist( z, D \ D ) > , dist( w, D \ D ) >
0, let Q D ( z, w ; D )denote the probability that a Brownian excursion in D from z to w staysin D . (A Brownian excursion in D is a Brownian motion starting at z andconditioned to go immediately into D and exit at w . It is not difficult tomake this precise.) We note that Q D ( z, w ; D ) is invariant under conformaltransformations of D , and if z, w are ∂D -analytic, Q D ( z, w ; D ) = H ∂D ( z, w ) H ∂D ( z, w ) . If D ⊂ H is simply connected with H \ D bounded and dist(0 , H \ D ) > Q H (0 , ∞ ; D ) = Φ ′ D (0) , where Φ D : D → H is a conformal transformation with Φ( z ) ∼ z as z → ∞ .When studying SLE it is useful to consider subdomains of H and the boundarypoint infinity. In order to make a number of formulas work in this case, it is useful toadapt the following “abuse of notation” about derivatives. This can be considereda kind of normalization at infinity. • When we consider the conformal transformation g : H → H given by g ( z ) = − /z , then we write(7) g ′ (0) = g ′ ( ∞ ) = − . • If D ⊂ H and H \ D is bounded, then we say that ∞ is ∂D -analytic. If D , D are two such domains and f : D → D is a conformal transforma-tion with f ( ∞ ) = ∞ , we define f ′ ( ∞ ) by f ( z ) ∼ zf ′ ( ∞ ) , z → ∞ . Equivalently, if F ( z ) = − /f ( − /z ) = g ◦ f ◦ g ( z ), then f ′ ( ∞ ) = F ′ (0) . • More generally, if F : D → D ′ is a conformal transformation with F ( z ) = ∞ or F ( ∞ ) = z , we compute derivatives using the chain rule and (7).The boundary Poisson kernel H ∂D ( z, w ) can be defined if z or w equals infinityusing the scaling rule (3). Under our normalization H ∂ H ( x, ∞ ) = 1. ♣ If D, D ′ are simply connected domains, z, w are distinct ∂D -analytic points, and z ′ , w ′ aredistinct ∂D ′ -analytic points, then there is a one parameter family of conformal transformations f : D → D ′ with f ( z ) = z ′ , f ( w ) = w ′ . The quantity f ′ ( z ) f ′ ( w ) is invariant of the choiceof the transformation. Our definitions of derivatives at infinity are made so that this propertyholds as well when w = ∞ or w ′ = ∞ . The annulus.
The functions that arise from the Poisson kernel of the annuluswill be important. By considering different winding numbers, using the scaling rule,and applying (6), we can see that H ∂A r (1 , e − r + ix ) = e r ∞ X k = −∞ H ∂S r (0 , x + ir ) = e r J ( r, x ) , where J ( r, x ) is defined by(8) J ( r, x ) = π r ∞ X k = −∞ (cid:20) cosh (cid:18) π ( x + 2 kπ )2 r (cid:19)(cid:21) − . We will view J ( r, x ) as a function on (0 , ∞ ) × R satisfying J ( r, x ) = J ( r, x + 2 π ).Under our normalization of the Poisson kernel,(9) e − r Z π H ∂A r (1 , e − r + ix ) dx = πr , which implies Z π J ( r, x ) dx = 2 Z π J ( r, x ) dx = 2 πr . Indeed, ( r/ π ) J ( r, x ) has the interpretation as the density of the angle of the hittingpoint of an h -process in A r started at 1 conditioned to leave A r at C r (in otherwords, the h -process associated to the harmonic function h ( z ) = − log | z | ). Usingthis interpretation, we can see that there exists ρ > r sufficientlysmall(10) ρ πr ≤ Z r J ( r, x ) dx ≤ (1 − ρ ) πr . ♣ To see (9) , recall that under our normalization of the Poisson kernel e − r Z π H A r ( e − ǫ , e − r + ix ) dx is π times the probability that a Brownian motion starting at e − ǫ = 1 − ǫ + O ( ǫ ) leaves A r at C r . A standard estimate for Brownian motion tells us that this probability equals ǫ/r . LE IN MULTIPLY CONNECTED DOMAINS 11
Lemma 3.1.
There exist c < ∞ such that if r ≥ , x ∈ R , (cid:12)(cid:12)(cid:12)(cid:12) J ( r, x ) − r (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − r . Proof.
We will assume r ≥ ≤ r ≤ V be a subset of[0 , π ) which can also be viewed as a periodic subset of R . We need to show that12 π Z V J ( r, x ) dx = l ( V ) r [1 + O ( re − r )] , where l denotes length. By definition,12 π Z V J ( r, x ) dx = e − r π Z V H ∂A r (1 , e − r + ix ) dx = e − r π Z V H ∂A r ( e − r , e ix ) dx. Let B t denote a complex Brownian motion and T s = inf { t : B t ∈ C s } . Let p ( z ; V ) = P z { B T ∈ V } , q ( z ; V ) = P z { B T ∈ V | T < T r } , and let q ± ( r ; V ) be the maximum and minimum of q ( z, V ) on C r − . Then, q − ( r, V ) ≤ re − r π Z V H ∂A r ( e − r , e ix ) dx ≤ q + ( r, V ) . Hence it suffices to show that if z ∈ C r − , q ( z, V ) = l ( V ) [1 + O ( r e − r )] , where the error term is uniform in z . If z ∈ C r − , then P z { T < T r } = 1 /r , andhence p ( z, V ) = r − q ( z, V ) + (1 − r − ) P z { B T ∈ V | T r < T } . Using the strong Markov property and the exact form of the Poisson kernel in thedisk, we see that p ( z, V ) = l ( V ) [1 + O ( | z | )] , P z { B T ∈ V | T r < T } = l ( V ) [1 + O ( | z | )] , and hence if z ∈ C r − , r − q ( r, V ) = l ( V ) [1 + O ( | z | )] − (1 − r − ) l ( V ) [1 + O ( | z | )]= l ( V ) [ r − + O ( e − r )]= r − l ( V ) [1 + O ( re − r )] . (cid:3) Another important function will be H I ( r, x ) = − xr + Z x J ( r, y ) dy = Z x (cid:20) J ( r, y ) − r (cid:21) dy, which satisfies H I ( r, x ) = H I ( r, x + 2 π ) and H ′ I ( r, x ) = J ( r, x ) − r . Here we are using the notation from [25], and the prime denotes an x -derivative. Lemma 3.2.
Let K ( r, x ) = r H I ( r, x ) . Then for all r , K is an odd function ofperiod π satisfying K ( r, π − x ) = K ( r, π + x ) , K (0) = K ( π ) = 0 , and K ( r, x ) ≤ π − x, ≤ x ≤ π. Moreover, there exists ǫ > such that for all r sufficiently small and all x , K ( r, x ) ≤ π − ǫr. Proof.
This is straightforward. The last estimate uses (10). (cid:3) ♣ Although we will not need it for our main theorem, in a comment in Section 7.1 we willuse the fact that the function Φ( r, x ) = r J ( r, x ) = r H ′ I ( r, x ) + 1 satisfies the differentialequation (11) ˙Φ = Φ ′′ + H I Φ + H ′ I Φ . Here, as later in the paper, we use dots for r -derivatives and primes for x -derivatives. To seethis, we will need the following fact from [25]: ˙ H I = H ′′ I + H ′ I H I . Hence G = H ′ I satisfies ˙ G = G ′′ + H I G ′ + H ′ I G, and ˙Φ = G + r G ′′ + r H I G ′ + r H ′ I G = G + Φ ′′ + H I Φ ′ + H ′ I (Φ − ′′ + H I Φ ′ + H ′ I Φ Lemma 3.3.
There exists c > such that the following holds. Suppose r ≥ and f : D → A r is a conformal transformation with f ( C ) = C where D = D \ K and K is a compact subset containing the origin. Then for | z | = 1 , | | f ′ ( z ) | − | ≤ c e − r , | f ′′ ( z ) | ≤ ce − r . Proof.
Let φ D be the harmonic function on D with boundary values 0 on C and1 on K and let φ r = φ A r . By conformal invariance, φ D ( z ) = φ r ( f ( z )) = − log | f ( z ) | r . Since f maps C to C , this implies r ∂ n φ D ( z ) = | f ′ ( z ) | , where n denotes the inward unit normal. Also, conformal invariance of excursionmeasure gives Z C ∂ n φ D ( z ) | dz | = Z C ∂ n φ r ( z ) | dz | = 2 πr . Hence to prove the first estimate, it suffices to show for z, w ∈ C ,(12) ∂ n φ D ( z ) = ∂ n φ D ( w ) [1 + O ( e − r )] . Using Koebe estimates, we can find a universal s such that for r sufficiently large, K ⊂ D r − s . Suppose we start Brownian motions at e − ǫ z and e − ǫ w , respectively.The probability that they reach C r − s without hitting C is ǫ/ ( r − s ). On C r − s , LE IN MULTIPLY CONNECTED DOMAINS 13 φ D ) = 1 − O ( r − ). Using Lemma 3.1, we can see that the conditional distributionson C r − s given that the Brownian motions reach C r − s are the same for z, w up to anerror of order O ( re − r ). Hence, if q ( z ) = q ( z, r, ǫ ) denotes the probability that theBrownian motion starting at z reaches C r − s before C but does not hit K before C , then | q ( z ) − q ( w ) | ≤ c ǫr − s r − O ( re − r ) ≤ ǫr O ( e − r ) , from which we conclude (12). Indeed, we conclude the stronger fact, q ( z ) = − log | z | r [1 + O ( e − r )] , e − < | z | < . This implies | f ( z ) | = | z | [[1 + O ( e − r )] , e − < | z | < . For the second estimate, fix z and assume without loss of generality that z = 1and f (1) = 1. By Schwarz reflection, we can extend f to a neighborhood of radius1 / , L ( z ) = log z, g ( z ) = log f . where L (1) = g (1) = 0. We have | Re g ( z ) − Re L ( z ) | ≤ e − r and g (1) = L (1). From this we can use standard argumentsto conclude that | g ( z ) − L ( z ) | = O ( e − r ). Using the Cauchy integral formula, weget | g ′ ( z ) − L ′ ( z ) | , | g ′′ ( z ) − L ′′ ( z ) | ≤ c O ( e − r ). (cid:3) A computation that we will do a little later will give us a particular annulusfunction A ( r, x ) which we now define. Suppose that D = S r , z = 0 , w = x + ri andlet γ t be a curve starting at the origin parametrized so that hcap[ γ t ] = t . Let D t be the domain obtained by splitting H by the nontrivial 2 πk translates of γ t , D t = S r \ [ k ∈ Z \{ } [ γ t + 2 πk ] , and let Q t = Q D (0 , w ; D t ) . Then (see the end of Section 3.8), one can check that as t → Q t = 1 − A ( r, x ) t + o ( t ) , where A ( r, x ) = X k ∈ Z \{ } H ∂S r (0 , πk ) H ∂S r (2 πk, x + ir ) H ∂S r (0 , x + ir ) . Using (5) and (6), we get(14) A ( r, x ) = π r X k ∈ Z \{ } cosh ( πx/ r )sinh ( π k/r ) cosh ( π ( x − πk ) / r ) , Proposition 3.4.
For fixed r , A ( r, · ) is a positive, even function, that is increasingin | x | . There exists c > such that If < r ≤ and ≤ x ≤ π , (15) A ( r, x ) ≤ cr exp (cid:26) − πr ( π − x ) (cid:27) . Proof.
The definition implies A ( r, x ) = A ( r, − x ). For r ≤
1, 0 ≤ x < π ,cosh ( πx/ r ) ≍ e πx/r , sinh ( π k/r ) cosh ( π ( x − πk ) / r ) ≍ e | k | π /r e π | πk − x | /r ≥ e | k | π /r e π (2 π − x ) /r . By summing over k , we get (15). The monotonicity in | x | will follow if we showthat that for each integer k ,cosh ( πx/ r )cosh ( π ( x − πk ) / r ) + cosh ( πx/ r )cosh ( π ( x + 2 πk ) / r )is an increasing function of | x | .Indeed, we will now show that if y ∈ R and f ( x ) = cosh x cosh ( x − y ) + cosh x cosh ( x + y ) , then f is increasing for x ≥
0. Since f ( x ) = cosh(2 x ) + 1cosh(2 x − y ) + 1 + cosh(2 x ) + 1cosh(2 x + 2 y ) + 1 , it suffices to show for every y ∈ R , that F ( x ) = cosh x + 1cosh( x − y ) + 1 + cosh x + 1cosh( x + y ) + 1 , is increasing for x ≥
0. Using the sum rule, we getcosh( x − y ) + 1 + cosh( x + y ) + 1 = 2 cosh x cosh y + 2 , Letting r = cosh y ≥
1, we get[cosh( x − y ) + 1] [cosh( x + y ) + 1] = (cosh x cosh y + 1) − sinh x sinh y = ( r cosh x + 1) − ( r − x − x + 2 r cosh x + r = (cosh x + r ) . Therefore, F ( x ) = 2 r (cosh x + r − ) (cosh x + 1)(cosh x + r ) = 2 r e G (cosh x ) , where G ( t ) = log( t + 1 r ) + log( t + 1) − t + r ) . Since r ≥ G ′ ( t ) > t > G and F are increasing. (cid:3) SLE κ in H . If κ = 2 /a ∈ (0 , chordal SLE κ (in H from to ∞ ) is thesolution to the chordal Loewner equation (16) ∂ t g t ( z ) = ag t ( z ) − U t , g ( z ) = z, where U t = − B t is a standard Brownian motion. With probability one [20], thisgenerates a random path γ : (0 , ∞ ) → H such that the domain of g t is H \ γ t . Thecurve is parametrized so that hcap[ γ t ] = at (see [10, Chapter 3] for definitions); inother words, g t ( z ) = z + atz + O ( | z | − ) , z → ∞ . LE IN MULTIPLY CONNECTED DOMAINS 15
For every r >
0, let τ r = inf { t > γ ( t ) S r } = inf { t > γ ( t ) = r } . ♣ SLE κ for κ > is also very interesting, but the paths are not simple. We restrict inthis paper to κ ≤ . Chordal
SLE κ produces a probability measure on curves, modulo (increasing)reparametrization, from 0 to ∞ . By conformal transformation, we get a probabilitymeasure on curves connecting distinct boundary points z, w of simply connecteddomains D . We will denote this measure by µ D ( z, w ). ♣ To get a measure on parametrized curves, one should use the natural parametrization asdescribed in [11]. This parametrization satisfies a conformal covariance rule under conformaltransformations. We would extend our definitions in this paper to parametrized curves, but itwould not add anything to our arguments here. For this reason we will consider curves moduloreparametrization as in [19].
Radial
SLE κ from to in D is defined by the transformations on the disk˜ g t ( e iz ) = e h t ( e iz ) where h t satisfies(17) ∂ t h t ( z ) = a ( h t ( z ) − U t ) , where, as in [25], we write cot ( z ) = cot( z/ U t is a standard Brownian mo-tion. By conformal invariance, this gives a probability measure on curves µ ( z, w )connecting one boundary point z and one interior point w .3.4. Brownian bubble measure.
Our main interest is the Brownian loop mea-sure. However, computations of the measure lead to considering excursions and theboundary bubble measure.Suppose D is a domain with smooth (not necessarily connected) boundary. Foreach z ∈ ∂D, V, V ⊂ ∂D , we define (Brownian) excursion measures by E D ( z, V ) = Z V H ∂D ( z, w ) | dw | , E D ( V , V ) = Z V E D ( z, V ) | dz | = Z V Z V H ∂D ( z, w ) | dz | | dw | . They satisfy the scaling rules E D ( z, V ) = | f ′ ( z ) | E f ( D ) ( f ( z ) , f ( V )) , E D ( V , V ) = E f ( D ) ( V , V ) . In particular E D ( V , V ) is a conformal invariant and hence is well defined even ifthe boundaries are not smooth. The quantity E D ( z, V ) needs local smoothness at z to be defined.Boundary bubbles in D are loops rooted at z ∈ ∂D and otherwise staying in D .We review the definitions (see [10, Section 5.5]). The bubble measure is a σ -finitemeasure on bubbles. In H we can define the measure, by specifying for each simplyconnected domain D ⊂ H with dist(0 , H \ D ) >
0, the measure of the set of bubblesat 0 that do not lie in D . Definition If D ⊂ H is a subdomain, x ∈ R , and dist( x, H \ D ) >
0, thenΓ( x ; D ) = Γ H ( x ; D ) = ∂ y [ H H ( z, x ) − H D ( z, x )] | z = x . The quantity Γ( x ; D ) is the bubble measure (in H rooted at x ) of bubbles thatintersect H \ D . Alternatively, we can write(18) Γ( x ; D ) = lim ǫ ↓ ǫ − E x + iǫ [ H ( B τ , x )] , where B is a complex Brownian motion and τ = τ D = inf { t : B t D } . Note thatΓ( x ; D ) − Γ( x ; D ) = ∂ y [ H D ( z, x ) − H D ( z, x )] | z = x . We can similarly define Γ D ( z ; D ′ ) if z is ∂D -analytic, D ′ ⊂ D , and dist( z, D \ D ′ ) >
0. It is defined as in (18), which we can also write asΓ D ( z ; D ′ ) = Z D ∩ ∂D ′ H D ( w, z ) d E D ( z, w ) . It satisfies the following scaling rule: if f : D → f ( D ) is a conformal transformation,then Γ D ( z ; D ′ ) = | f ′ ( z ) | Γ f ( D ) ( f ( z ) , f ( D ′ )) . If D ⊂ H is simply connected, this quantity can be computed [10, Proposition 5.22]:if f : D → H is a conformal transformation with f ( x ) = x , then(19) Γ( x ; D ) = − Sf ( x ) , where S denotes the Schwarzian derivative. Particular cases of importance to usare considered in the following proposition. Proposition 3.5. Γ H (0; S r ) = π r , If Γ( r ) = Γ D (1; A r ) , then as r → ∞ , (20) Γ( r ) = 1 + O ( e − r )2 r . Moreover, Γ( r ) = π r + δ ( r ) , where (21) δ ( r ) = X k ∈ Z \{ } [ H ∂ H (0 , πk ) − H ∂S r (0 , πk )] = 112 − π r ∞ X k =1 (cid:20) sinh (cid:18) kπ r (cid:19)(cid:21) − . In particular, δ ( r ) = 1 + O ( e − r )2 r − π r . Proof.
Since S r is simply connected, we can use (19) with f ( z ) = e πz/r − H (0; S r ) = − Sf (0)6 = π r . The second equality follows from (18). Indeed, as noted previouslylim ǫ ↓ ǫ − P − ǫ { B τ ∈ C r } = 1 /r, LE IN MULTIPLY CONNECTED DOMAINS 17 and the exact form of Poisson kernel in D shows that H D ( z,
1) = 12 + O ( e − r ) , z ∈ C r . To each Brownian bubble in D rooted at 1 that intersects C r , there is a corre-sponding path in H that starts at 0, ends at 2 πk for some integer k , and does notstay in S r . Only those paths that end at 0 are Brownian bubbles in H rooted at 0.Therefore, to compute Γ H (1; S r ) we can subtract the measure of the other bubbles.To get the measures of the bubbles to be subtracted we consider the measure ofexcursions in H minus the measure of excursions in S r . We therefore getΓ H (0; S r ) = Γ D (1; A r ) − X k ∈ Z \{ } [ H ∂ H (0 , πk ) − H ∂S r (0 , πk )]= Γ D (1; A r ) − δ ( r ) . Using (5) and H ∂ H (0 , x ) = x − , we see that δ ( r ) = 2 ∞ X k =1 πk ) − π r (cid:20) sinh (cid:18) kπ r (cid:19)(cid:21) − ! = 112 − π r ∞ X k =1 (cid:20) sinh (cid:18) kπ r (cid:19)(cid:21) − . (cid:3) Brownian loop measure.
In order to describe
SLE κ in other domains, weintroduce the Brownian loop measure as first introduced in [18]. Definition
The rooted Brownian loop measure on C is the measure on loops givenby(22) 12 πt dt × area × ν BB , where ν BB denotes the probability measure induced by a Brownian bridge of timeduration one at the origin.To be more precise, a rooted loop is a continuous function η : [0 , t η ] → C with η (0) = η ( t η ). Such a loop can be described by a triple ( t, z, ¯ η ) where t > z = η (0) is the root, and ¯ η : [0 , → C is a loop of time durationone starting at the origin. The rooted loop measure is obtained by choosing ( t, z, ¯ η )according to the measure (22). If D ⊂ C , the the rooted loop measure in D is therooted loop measure in C restricted to loops that lie in D . Definition
The rooted loop measure on a domain D induces a measure on unrootedloops which we denote by m D . We consider this as a measure on unrooted loopsmodulo reparametrization. (However, the proof of conformal invariance requiresconsidering the parametrized loops.)For the purposes of this paper, we do not need to worry about the time parametriza-tion of the loops. The fundamental fact that explains the importance of the loopmeasure is the following. We do this to emphasize that we do not need to assumethat D is simply connected. Proposition 3.6 (Conformal invariance) . If f : D → f ( D ) is a conformal trans-formation, then f ◦ m D = m f ( D ) . ♣ We have stated the proposition for loops, modulo reparametrization. One can get asimilar result for parametrized loops but then one must change the parametrization as in theconformal invariance of Brownian motion.
Sketch of proof.
Let ρ ( z, z ; t ) be the measure on paths associated to Brownian loopsat z of time duration t . It is a measure of total mass p t ( z, z ) = (2 πt ) − that canbe defined using standard Brownian bridge techniques. Let ρ ( z, z ) = Z ∞ ρ ( z, z ; t ) dt, which is an infinite measure. For any D , we define ρ D ( z, z ; t ) , ρ D ( z, z ) by restriction.If f : D → f ( D ) is a conformal transformation, and η is a loop in D , we write f ◦ η for the corresponding loop in f ( D ) obtained using Brownian scaling on theparametrization. In other words, if η has time duration t η , then f ◦ η has timeduration Z t η | f ′ ( η ( s )) | ds. The measure ρ D ( z, z ) induces a measure f ◦ ρ D ( z, z ) by considering f ◦ η . Usingthe conformal invariance of Brownian motion, one can check that(23) f ◦ ρ D ( z, z ) = ρ f ( D ) ( f ( z ) , f ( z )) . Suppose h is a continuous, nonnegative function on D . Then h induces a measureon (rooted) loops by ρ D,h = Z D ρ D ( z, z ) h ( z ) dA ( z ) , where A denotes area. We can also consider this as a measure on unrooted loopsby forgetting the root. We write ρ D for ρ D,h with h ≡
1. Another way to definethe Brownian loop measure µ D on unrooted loops is dµ D dρ D ( η ) = 1 t η , where t γ denotes the time duration of γ . More generally, dµ D dρ D,h ( η ) = (cid:20)Z t η h ( η ( s )) ds (cid:21) − . Suppose h ( z ) = | f ′ ( z ) | . Then (23) implies that f ◦ ρ D,h = Z D ρ f ( D ) ( f ( z ) , f ( z )) | f ′ ( z ) | dA ( z )= Z f ( D ) ρ f ( D ) ( w, w ) dA ( w ) = ρ f ( D ) . Also, Z t η h ( η ( s )) ds = Z t η | f ′ ( η ( s )) | ds = 1 t f ◦ η . (cid:3) By construction, m D also satisfies the restriction property. LE IN MULTIPLY CONNECTED DOMAINS 19 • If V , V are subsets, we write either m ( V , V ; D ) or m D ( V , V ) for the m D measure of the set of loops in D that intersect both V and V . • Suppose D ⊂ H is a domain (not necessarily simply connected) withdist(0 , H \ D ) >
0. Suppose γ satisfies (16) and t < T := inf { t : dist( γ ( t ) , H \ D ) = 0 } . Then(24) m ( γ t , H \ D ; H ) = a Z t Γ( U s ; g s ( D )) ds. If D is simply connected, we can use (19) to write(25) m ( γ t , H \ D ; H ) = − a Z t Sf s ( U s ) ds, where f s is a conformal transformation of g s ( D ) onto H with f ( U s ) ∈ R . ♣ The only functionals of the Brownian loop measure that we will need are of the typeon the left-hand side of (24) . We might consider using the right-hand side of (24) as the definition of m ( γ t , H \ D ; H ) . However, it is not so easy to see from this formulation to seethat if γ is a curve in H connecting boundary points , x , then m ( γ t , H \ D ; H ) = m ( γ Rt , H \ D ; H ) , where γ Rt denotes the reversal of the path. This is immediate from the loop measure descriptionof the quantity. ♣ The formula (24) comes from a Brownian bubble analysis of the Brownian loop measure.Suppose γ is a simple curve from to ∞ in H . If l is a loop in H that intersects γ , we canconsider the first time (using the time scale of γ ) that the loop intersects γ . If l intersects γ first at time t , then l is a “boundary bubble” in H \ γ t rooted at γ ( t ) . We therefore can writethe Brownian loop measure, restricted to loops intersecting γ , as an integral of the Brownianbubble measure in decreasing family of domains H \ γ t . We can think of γ as an “explorationprocess” for the Brownian loop measure. This idea is used in the construction of conformalloop ensembles by Sheffield and Werner [22]. This exploration idea is important in our analysisof SLE κ in an annulus. Although the Brownian loop measure is a measure on unrooted loops, it is oftenconvenient to choose roots of the loops. For example, if η is an unrooted loop, wecan choose the root to be the closest point to the origin, say e − r + iθ . (Except for aset of measure zero, this point will be unique). The rooted loop is then a Brownianbubble in the domain O r := C \ D r . This is the basis for the following computation. Proposition 3.7.
Suppose D ⊂ D is a simply connected domain with dist(0 , ∂D ) >e − r . Then, m ( D r , D \ D ; D ) = 1 π Z ∞ r Z π Γ O s ( e − s + iθ ; D ) ds dθ, where O s = D \ D s . Lemma 3.8.
There exists c < ∞ such that the following is true. Suppose D ⊂ D is a simply connected domain containing the origin and g : D → D is the conformaltransformation with g (0) = 0 , g ′ (0) > . Suppose that r ≥ log g ′ (0) + 2 . Let φ : g ( A r ∩ D ) → A s be a conformal transformation sending C to C and let h = φ ◦ g which maps A r ∩ D onto A s . Then if u = r − log g ′ (0) , z ∈ C r , w ∈ C , | s − u | ≤ c e − u , | φ ′ ( w ) − | ≤ c e − u , | h ′ ( z ) − g ′ (0) | ≤ c g ′ (0) e − u , (cid:12)(cid:12) m D ( D r , D \ D ) − log( r/u ) (cid:12)(cid:12) ≤ c e − u . Proof.
The Koebe-1 / g − shows that dist(0 , ∂D ) ≥ [4 g ′ (0)] − . Applying the distortion theorem to g restricted to D u + , we see that there exists c < ∞ such that if | w | ≤ e − r , | g ( w ) − g ′ (0) w | ≤ c e − u , | g ′ ( w ) − g ′ (0) | ≤ c e − u . In particular, if | w | = e − r , then(26) | g ( w ) | = e − u [1 + O ( e − u )] . Using this and monotonicity, we see that(27) s = u + O ( e − u ) . Let U denote the conformal annulus g ( A r ∩ D ) so that φ maps U onto the annulus A s . By conformal invariance we see that log | g ( z ) | /s equals the probability that aBrownian motion starting at z exits U at g ( C r ). However, we know that the innerboundary of U lies within distance O ( e − u ) of C u . If the Brownian motion getsthat close to C u , the probability that it does not exit at C u is O ( e − u /u ). Therefore,log | g ( z ) | s = log | z | u [1 + O ( e − u /u )] . Hence, from (27), we get log | g ( z ) | = log | z | [1 + O ( e − u )] , which implies | g ′ ( e iθ ) | = 1 + O ( e − u ). The argument to show that | h ′ ( e − r + iθ ) | = g ′ (0)[1 + O ( e − u )] is similar.By conformal invariance and symmetry, E A r ∩ D ( C r , ∂D ) = E A s ( C s , C ) = E A s ( C , C s ) = 2 πs − . Similarly, if ˆ v ( z ) = P z { B ˆ σ ∈ C } = 1 − log | z | r , where ˆ σ = inf { t : B t A r } , then2 πr − = E A r ( C r , C ) = Z C r ∂ n ˆ v ( z ) | dz | . By the strong Markov property, we can write E A r ( C r , C ) = Z D ∩ ∂D (cid:20) − log | z | r (cid:21) d E A r ( C r , dz ) . LE IN MULTIPLY CONNECTED DOMAINS 21
The term 1 − log | z | r is the probability that a Brownian motion starting at z exits A r at C . Therefore, using (27), Z D ∩ ∂D log | z | r d E A r ( C , dz ) = 2 π [ s − − r − ] = 2 π [ u − − r − + O ( e − u /u )] . Lemma 3.1 implies that if V ⊂ ∂D and z, w ∈ C r , E A r ( z, V ) = E A r ( w, V ) [1 + O ( e − u )] , and hence E A r ( z, V ) = 12 π E A r ( C , V ) [1 + O ( e − u )] . Lemma 3.1 can also be used to see that if w ∈ D ∩ ∂D, z ∈ C r , H A r ( w, C r ) = 12 log | w | r [1 + O ( e − u )] . Therefore, using (27),Γ A r ( z, A r ∩ D ) = Z D ∩ ∂D H A r ( w, z ) d E A r ∩ D ( z, w ) = u − − r − O ( e − u )] . From Proposition 3.7 we know that the quantity we are interested in can bewritten as1 π Z π Z ∞ r Γ A t ( e − t + iθ ; A t ∩ D ) dt dθ = Z ∞ (cid:20) u + t − r + t (cid:21) [1 + O ( e − u − t )] dt. By computing the integral we see that this quantity equalslog( r/u ) + O ( e − u ) . (cid:3) We will need to consider the Brownian loop measure in an annulus. If we fixthe origin as a marked point, we can divide loops into two sets: those with nonzerowinding number around zero and those with zero winding number. If A is a confor-mal annulus such that 0 and ∞ lie in different components of A c , then the measureof the set of loops in A with nonzero winding number is finite. It is a conformalinvariant which we calculate in the next proposition. Proposition 3.9.
Let m ∗ ( r ) denote the Brownian loop measure of loops in A r thathave nonzero winding number. Then m ∗ ( r ) = r − Z r δ ( s ) ds, where δ ( s ) is defined as in (21) . In particular, there exists C > such that as r → ∞ , (28) e m ∗ ( r ) = C r − e r/ [1 + O ( r − )] . Proof.
By focusing on the point of the loop of largest radius (see the appendix of[14]), we can give the expression m ∗ ( r ) = 2 π Z r X k ∈ Z \{ } π H S s (0 , πk ) = 16 − δ ( r ) . Proposition 3.5 implies that there exists c such that m ∗ ( r ) = r − log r + c + O ( r − ) , r → ∞ , from which (28) follows with C = e c . (cid:3) Corollary 3.10. • Suppose D ⊂ D is a simply connected domain containing the origin andsuppose that dist(0 , ∂D ) > e − r . Let ≤ s < r be defined by saying that theannulus D \ D r is conformally equivalent to A s . Then the Brownian loopmeasure of loops in A r of nonzero winding number that intersect D \ D is m ∗ ( r ) − m ∗ ( s ) . • Under the same assumptions, the Brownian loop measure of loops in D ofnonzero winding number that intersect D \ D is log g ′ (0) / where g : D → D is the conformal transformation with g (0) = 0 , g ′ (0) > .Proof. The first assertion follows immediately and the second is obtained by con-sidering comparing D \ A r and D \ A r as r → ∞ . (cid:3) We ill use the following estimate in the discussion in the next section but it willnot figure in our main results. See [ ? ] for a proof. Proposition 3.11.
Let k ( r ) denote the m D − r measure of loops that intersect both A − r \ A − r +1 and D . Let k ′ ( r ) be the measure of such loops that do not separate theorigin from C . Then as r → ∞ , k ( r ) = r − + O ( r − ) , k ′ ( r ) = O ( r − ) . In particular, if V , V are disjoint compact sets, then there exists Λ( V , V ) suchthat as r → ∞ , m D − r ( V , V ) = log r − Λ( V , V ) + o (1) . Chordal
SLE κ in simply connected domains. We will review two equiv-alent ways to construct
SLE κ in simply connected domains for κ = 2 /a ≤
4. See[12, 18, 10, 9] for more details. Suppose D is a simply connected subdomain of H with dist(0 , H \ D ) >
0. Let w be a nonzero ∂D -analytic point; we allow w = ∞ as a possibility. Let Φ : D → H be the unique conformal transformation withΦ(0) = 0 , Φ( w ) = ∞ , | Φ ′ ( w ) | = 1. Here we are using the conventions about deriva-tives as discussed in Section 3.1. The most important example for this paper is D = S r and w = x + ir for some x ∈ R .Let g t be the solution of the Loewner equation ∂ t g t ( z ) = ag t ( z ) − U t , g ( z ) = z, where U t = − B t is a standard Brownian motion defined on the probability space(Ω , F , P ). Then the corresponding curve γ is SLE κ in H from 0 to ∞ which with P -probability one is a simple curve with γ (0 , ∞ ) ⊂ H .Let T = T D = inf { t > γ ( t ) D } . For t < T , let w t = g t ( w ) , γ ∗ t = Φ ◦ γ t , and let ˆ g t be the unique conformal transformation of H \ γ ∗ t onto H with ˆ g t ( z ) = z + o (1) as z → ∞ . Let Φ t = ˆ g t ◦ Φ ◦ g − t . LE IN MULTIPLY CONNECTED DOMAINS 23
Then ˆ g t satisfies the Loewner equation ∂ t ˆ g t ( z ) = a Φ ′ t ( U t ) ˆ g t ( z ) − ˆ U t , ˆ g ( z ) = z, where ˆ U t = ˆ g t ( γ ∗ ( t )) = Φ t ( U t ). Then Φ t is the unique conformal transformaton of g t ( D \ γ t ) onto H with Φ t ( U t ) = ˆ U t , Φ t ( w t ) = ∞ , | Φ ′ t ( w t ) | = | g ′ t ( w ) | − . Moreover,using only the Loewner equation, one can show that(29) ˙Φ t ( U t ) = − b ′′ t ( U t ) , ˙Φ ′ t ( U t ) = a Φ ′′ t ( U t ) ′ t ( U t ) − a Φ ′′′ t ( U t )3where ˙Φ t ( U t ) , ˙Φ ′ t ( U t ) denote ∂ t Φ t ( x ) , ∂ t Φ ′ t ( x ) evaluated at x = U t .Let(30) H t = H ∂g t ( D \ γ t ) ( x, w t ) , K t = | g ′ t ( w ) | b H t ( U t ) b = Φ ′ t ( U t ) b . The second equality for K t follows from the scaling rule for the Poisson kernel. Astraightforward Itˆo’s formula calculation using (29) shows that dK t = K t (cid:20) a c S Φ t ( U t ) dt + b H ′ t ( U t ) H t ( U t ) dU t (cid:21) , where S denotes the Schwarzian derivative. Let M t = exp (cid:26) − a c Z t S Φ s ( U s ) ds (cid:27) K t = exp n c m H ( γ t , H \ D ) o | g ′ t ( w ) | b H t ( U t ) b . (To check the second equality, recall that we have parametrized so that hcap( γ t ) = at .) Then M t is a local martingale satisfying dM t = b H ′ t ( U t ) H t ( U t ) M t dU t = b Φ ′′ t ( U t )Φ ′ t ( U t ) M t dU t . We can use Girsanov theorem to define a new probability measure P ∗ obtainedby weighting by the local martingale M t . (The Girsanov theorem is stated fornonnegative martingales; since we only have a local martingale, we need to usestopping times. However, as long as t < T , there is no problem.) The Girsanovtheorem states that(31) dU t = b H ′ t ( U t ) H t ( U t ) dt + dW t , t < T, where W t is a standard Brownian motion with respect to P ∗ .Another application of Itˆo’s formula using (29) shows that if U t satisfies (31),then ˆ U t = Φ t ( U t ) satisfies d ˆ U t = Φ ′ t ( U t ) dW t . The upshot is that, with respect to the measure P ∗ , η t has the distribution of (atime change of) SLE κ from 0 to ∞ in H . Since γ t = Φ − ◦ η t , this implies that withrespect to P ∗ , γ t has the distribution of SLE κ from 0 to w in D . The Girsanovtransformation (31) is sufficent for understanding the probability measure µ D (0 , w ).Note that it is determined by the logarithmic derivative of H t ; the “compensator”terms do not need to be computed.The example of importance in this paper is D = S r and w = x + ir . It willsuffice for us to consider the probability measure µ S r (0 , w ). The drift term in (31) is somewhat complicated to write down; however, at time t = 0, we can use (6) tosee that it equals b L ( r, x ) where(32) L ( r, x ) = H ′ ∂S r (0 , x + ir ) H ∂S r (0 , x + ir ) = πr tanh (cid:16) πx r (cid:17) , where the prime denotes derivative in the first component. This measure is thesame (modulo time change) as the conformal image of SLE κ from 0 to ∞ in H ; inparticular, with probability one, the path leaves S r at w .In analyzing annulus SLE κ we will be studying measures that will turn out to beabsolutely continuous with respect to µ S r (0 , x + ir ). To review the issues that weneed to address, let us recall the case of SLE κ from 0 to ∞ in a simply connecteddomain D with H \ D bounded and dist(0 , H \ D ) >
0. In this case, when we weightby the appropriate local martingale M t , then with P ∗ -probability one, T = ∞ and γ ( t ) → ∞ . If T = ∞ and γ ( t ) → ∞ , then a deterministic estiamte gives M ∞ = exp n c m H ( γ, H \ γ ) o { γ ⊂ D } , and since this happens with P ∗ -probability one,(33) E [ M ∞ ] = M = Φ ′ (0) b . ♣ The argument we will use for the annulus is similar to the proof for simply connecteddomains, so it is worth reviewing the main steps. Suppose D is a simply connected domainwith H \ D bounded and w = ∞ . Here we were able to guess the exact form for the partitionfunction for µ D (0 , ∞ ) , Φ ′ D (0) b . Direct Itˆo’s formula calculation shows that M t as above givesa local martingale. However, to justify (33) , we need that fact that the curve weighted bythe local martingale goes to infinity without leaving the domain. This gives the necessary“uniform integrability”.In the annulus case, we will consider two measures on curves from to w = x + ir in S r . Wewill use the Feynman-Kac theorem applied to a slightly different process to give a candidatefor the partition function. Although we will not have an explicit form of it, we will know thatit satisfies a certain PDE and hence gives us a local martingale. Having a local martingaleis not sufficient; we will also need to show that the process weighted by the local martingaleleaves the domain at w . This will give the analogue of (33) . The argument for the annulus,as well as the argument here, will require κ ≤ . Shrinking domains.
We will need a generalization of this where the domain D is replaced with a decreasing family of domains { D t : t > } . Although whatwe describe can be done more generally, we will restrict to the case that we needin this paper. This will lead to a process that we call locally chordal SLE κ in anannulus. Let D = S r and w ∈ ∂S r \ { } . (The case S ∞ = H , w = ∞ corresponds to radial SLE and is discussed in the next subsection.) Let(34) ˜ γ t = [ k ∈ Z \{ } ( γ t + 2 πk ) , D t = D \ ˜ γ t . and ˆ D t = D \ (˜ γ t ∪ γ t ) = ψ − [ D \ η t ] , LE IN MULTIPLY CONNECTED DOMAINS 25 where η t = ψ ◦ γ t . In other words, when we slit the domain D = S r by γ t we alsoadd slits at the 2 πk translates of γ t .Let T denote the first t > γ ( t ) ∈ ∂S r or η t disconnects theorigin from the unit circle, T = inf { t > γ t D t } . Let ˜ D t = g t ( ˆ D t ), and, as before, U t = g t ( γ ( t )). We want to study the process thatevolves at time t like chordal SLE κ from γ ( t ) to w in the domain D t . Equivalently,the process after conformal transformation by g t evolves like chordal SLE κ from U t to w t = g t ( w ) in ˜ D t . The latter process can be defined in two equivalent ways.Let H t ( x ) = H ∂g t ( D \ γ t ) ( x, w t ) as in the previous section and let˜ H t ( x ) = H ∂ ˜ D t ( x, w t ) , Q t ( x ) = ˜ H t ( x ) H t ( x ) . The process can be considered as either of the following. • SLE κ in H from 0 to ∞ weighted by ˜ H t ( U t ) b . • SLE κ in S r from 0 to w weighted by Q t ( U t ) b . ♣ If J t is a positive process, then “weighting by J t ” is in the sense of the Girsanov thoerem.If J t satisfies dJ t = J t [ R t dt + A t dU t ] . then N t := exp (cid:26) − Z t R s ds (cid:27) J t , is a local martingale satisfying dN t = A t N t dU t . When we use the Girsanov theorem (using stopping times so that the local martingale is amartingale), then dU t = A t dt + dW t , where W t is a Brownian motion in the new measure. Let(35) ∆ t = ˙ Q t ( U t ) Q t ( U t ) , where ˙ Q t ( U t ) denotes ∂ t Q t ( x ) evaluated at x = U t . Our assumptions allow us toconclude that ∆ t is well defined and continuous for t < T .As in (30), we define K t = | g ′ t ( w ) | b ˜ H t ( U t ) b = | g ′ t ( w ) | b H t ( U t ) b Q t ( U t ) b . Using the previous calculation and the chain rule, we see that K t satisfies dK t = K t "(cid:16) − b ∆ t + a c S Φ t ( U t ) (cid:17) dt + b ˜ H ′ t ( U t )˜ H t ( U t ) dU t . If C t = exp (cid:26)Z t ∆ s ds (cid:27) ,M t = C bt exp (cid:26) − a c Z t S Φ s ( U s ) ds (cid:27) K t , then M t is a local martingale satisfying dM t = b ˜ H ′ t ( U t )˜ H t ( U t ) M t dU t . The term − a Z t S Φ s ( U s ) ds can be interpreted in terms of Brownian loops, but we need to be careful. At time s , − S Φ s ( U s ) / H rooted at U s thatintersect g s ( D s ). For every Brownian loop l , let s ( l ) be the smallest s such that s ( l ) ∩ γ s = ∅ . Then − a Z t S Φ s ( U s ) ds = ˜ m t , where ˜ m t = log ˜Λ t is the Brownian loop measure of l in D with s ( l ) ≤ t and l ∩ D \ D s ( l ) = ∅ . Then the local martingale is M t = C bt ˜Λ c / t H t ( U t ) b Q bt = C bt ˜Λ c / t ˜ H t ( U t ) b . Note that the only term in M t that has nontrivial quadratic variation is ˜ H t ( U t ) b .Therefore, when we weight by the local martingale, the process looks locally like SLE κ from γ ( t ) to w in D t . We call it locally chordal SLE κ (we have defined it onlyfor κ ≤ S r . We willuse κ ≤ S r at w . We can alsoview the paths as living in the annulus A r and going from 1 to e − r + ix with a knowntotal winding number. In Section 3.9 we will use an annulus reparametrization ofthe curve.3.8. Radial
SLE κ raised to H . Suppose D is a simply connected domain, z ∈ ∂D , w ∈ D , and ∂D is locally analtyic at z . Radial SLE κ in D from z to w is a measureon paths µ D ( z, w ) = Ψ D ( z, w ) µ D ( z, w ) , that satisfies the conformal covariance rule f ◦ µ D ( z, w ) = | f ′ ( z ) | b | f ′ ( w ) | ˜ b µ f ( D ) ( f ( z ) , f ( w )) . The conformal covariance rule determine the total mass up to a multiplicativeconstant and for convenience we choose the constant so that Ψ D (1 ,
0) = 1.To obtain the probability measure µ D (0 , w ) where w ∈ H , we weight chordal SLE κ by a particular local martingale. Let g t be the conformal maps for chordal SLE κ from 0 to ∞ , and let w ∈ H . Let Z t = g t ( w ) − U t and M t = | g ′ t ( w ) | ˜ b H H ( Z t , U t ) b , where b, ˜ b are the boundary and interior scaling exponents, respectively, as in (1).Then M t is a local martingale and the measure on the paths obtained by weightingby this local martingale is that of radial SLE κ . In the weighted measure, the pathstops at finite (half plane capacity) time T w at which γ ( T w ) = w . This determinesthe probability measure µ H (0 , w ) and conformal invariance determines the measurefor all simply connected D . Although this is not the same definition as originallygiven by Schramm [19], the Girsanov theorem shows that it is equivalent.One can also understand the relationship between radial and chordal SLE κ usingthe Brownian loop measure. Suppose that γ t is a simple curve in H starting at the LE IN MULTIPLY CONNECTED DOMAINS 27 origin and let η t = ψ ◦ γ t . We will assume that t is small so that η t is also simple.Let ˜ h t : D \ η t → D be the conformal transformation with ˜ h ′ t (0) > h ′ t (0) = e t . Let g t : H \ γ t be the usual conformal transforma-tion with driving function U t ; one can show that ∂ t hcap[ γ t ] | t =0 = 2 , which is why this is a standard choice of parametrization for chordal SLE . Let˜ γ t , ˆ γ t be as in the previous subsection and let h t be a conformal transformation h t : H \ ˆ γ t → H such that ψ ( h t ( z )) = ˜ h t ( ψ ( z )). This transformation is determineduniquely by requiring that h t ( iy ) = i [ y − t ] + o (1) , y → ∞ . We define φ t by h t = φ t ◦ g t . Let µ , µ denote µ H (0 , ∞ ) and µ D (1 , γ t by pulling back by ψ . (Note that | ψ ′ (0) | = 1 so thederivative factor in the scaling rule equals one.) We view these measures on theinitial segment γ t . The measure µ is supported on curves such that γ t ∩ ˜ γ t = ∅ .Note that µ ≪ µ , and let Y t ( γ t ) denote the Radon-Nikodym derivative so that dµ = Y dµ . Let Ψ ∗ denote the partition function for the raised radial SLE ; inparticular, Ψ ∗ H (0 , ∞ ) = 1.Although the loop measure is conformally invariant, we must be careful herebecause ψ : H → D is not one-to-one. Indeed, each loop l ′ in D has an infinitenumber of preimages in H . If l ′ is a loop in D that intersects η t , we can specify aunique preimage by considering the smallest s such that η s ∈ l ′ and then rooting l ′ at η s . We associate to l ′ the corresponding loop l in H rooted at γ s .Also, the loops of nonzero winding number in D have preimages that are notloops in H . Since the paths have been parametrized so that ˜ h ′ (0) = e t , Corollary3.10 implies that the measure of such loops is deterministic and equal to t/
6. Usingthis idea, we get the formal expression Y ( γ t ) = C t exp n c m ( γ t ) − ( t/ o Ψ ∗ H \ ˆ γ t ( γ ( t ) , H \ γ t ( γ ( t ) , . Here C t is a normalization to make this a probability measure and ˆ m ( γ t ) denotesthe measure of loops l in H that intersect γ t with the following property. • Let s be the smallest time with γ t ∈ l . Then l ∩ ˜ γ s = ∅ . In other words, the loop hits a translate of γ t before it hits γ t where timeis measured on the curve γ t .The ratio of partition functions is only formal but we can make sense of it by writingΨ ∗ H \ ˆ γ t ( γ ( t ) , ∞ )Ψ H \ γ t ( γ ( t ) , ∞ ) = Ψ ∗ H \ ˆ γ t ( γ ( t ) , ∞ )Ψ H \ ˆ γ t ( γ ( t ) , ∞ ) Ψ H \ ˆ γ t ( γ ( t ) , ∞ )Ψ H \ γ t ( γ ( t ) , ∞ ) . The first term on the right equals one since, formally,Ψ ∗ H \ ˆ γ t ( γ ( t ) , ∞ )Ψ H \ ˆ γ t ( γ ( t ) , ∞ ) = | h ′ t ( γ ( t )) | b Ψ ∗ H ( h t ( γ ( t )) , ∞ ) | h ′ t ( γ ( t )) | b Ψ H ( h t ( γ ( t )) , ∞ ) = 1 . For the second term, we use the formal computationΨ H \ ˆ γ t ( γ ( t ) , ∞ )Ψ H \ γ t ( γ ( t ) , ∞ ) = | g ′ t ( γ ( t )) | b Ψ g t ( H \ ˆ γ t ) ( g t ( γ ( t )) , ∞ ) | g ′ t ( γ ( t )) | b Ψ H ( U t , ∞ ) = Ψ g t ( H \ ˆ γ t ) ( U t , ∞ ) , and conformal covariance, Ψ g t ( H \ ˆ γ t ) ( U t , ∞ ) = φ ′ t ( U t ) b . Therefore, Y t ( γ t ) = C t e − c t/ exp n c m ( γ t ) o φ ′ t ( U t ) b . This is a local martingale (and a martingale for κ ≤
4) for chordal
SLE κ andwhen we weight by the martingale we get locally chordal SLE κ from γ ( t ) to ∞ in H \ ˆ γ t . Although we are considering chordal SLE κ , we are using the radialparametrization. This is the same as radial SLE κ viewed on the covering space H .It remains to find the normalization factor C t . Since the weighted measure locallylooks like chordal SLE κ in the infinitely slit domain and hence after mapping by h t looks like chordal SLE κ , we get that C t = e ˜ bt for some ˜ b . To find the exponent weneed only differentiate at 0. The measure of loops that hit both γ t and a translateof γ t is of order t and hence ∂ t ˆ m ( γ t ) (cid:12)(cid:12) t =0 = 0 . We claim that(36) ∂ t φ ′ t ( U t ) (cid:12)(cid:12) t =0 = − , and hence ˜ b = c
12 + b . Let us sketch the proof of (36). We write “small error” for errors that are o ( t )as t ↓
0. The quantity φ ′ t ( U t ) is the probability that a Brownian excursion in H \ ˆ γ t from γ ( t ) to ∞ does not hit ˜ γ t . Up to small error, it is the probability that anexcursion in H from 0 to ∞ does not hit ˜ γ t . The set ˜ γ t is a union of curves ofhalf-plane capacity 2 t rooted at the points 2 πk , k ∈ Z \ { } . The probability thatan excursion hits the translate γ t + 2 πk is exactly ∂ y q ( iy )where q ( z ) = E z [Im[ B τ ]], B is a standard Brownian motion and τ is the first timethat it leaves H \ [ γ t + 2 πk ]. As t ↓
0, up to small error this equals1(2 πk ) hcap[ γ t ] = t π . The probability of hitting more than one translate is O ( t ), and hence, up to smallerror, the probability that the excursion hits ˜ γ t is X k ∈ Z \{ } t π = t . LE IN MULTIPLY CONNECTED DOMAINS 29 ♣ In the last computation we use the fact that for a small curve rooted at x ∈ R , theexpected value of Im( B τ ) is given by the half-plane capacity times a multiplicative constantof the Poisson kernel. In order to keep track of constants (perhaps made especially confusingby our definition of H ), it is useful to remember that for large y if D = H \ D , E iy [Im( B τ )] ∼ y = H H ( y, . Hence, we get the general relation, E z [Im( B τ )] ∼ H H ( z, x ) hcap[ γ t ] . The estimate (13) is done similarly. In this case, the probability that an excursionfrom 0 to x + ir in S r hits the translate γ t + 2 πk is exactly, ∂ y q ( y ) where q ( z ) = E z [ H S r ( B τ , x + ir )] H ∂S r (0 , x + ir ) . Here τ is the first time that the Brownian motion leaves S r \ [ γ t + 2 πk ]. Up to smallerror, if B τ ∂S r , H S r ( B τ , x + ir ) = Im[ B τ ] H S r (2 πk, x + ir ) . Also, as y ↓ ∂ y E iy [Im( B τ )] | y =0 = hcap[ γ t ] H S r (0 , πk ) [1 + o (1)] . Annulus Loewner equation.
We will need to consider the annulus Loewnerequation which is similar to the chordal equation (16). We will need to define theannulus equation in the covering space S r . We start with some defintions. Assume U : [0 , ∞ ) −→ R is continuous with U = 0 and such that the chordal equation (16)produces a simple curve. Recall that ψ ( z ) = e iz , τ r = inf { t : Im γ ( t ) = r } , and let η t = ψ ◦ γ t . Let ˜ γ t = [ k ∈ Z \{ } ( γ t + 2 πk ) , ˆ γ t = γ t ∪ ˜ γ t ,T = inf { t : γ t ∩ ˜ γ t = ∅} . Equivalently, T is the first time that the curve η t is not simple. Note that T = τ r for each r ; indeed, by the definition of T , there must be an s < T with Im γ ( s ) =Im γ ( T ). Let S r,t = S r \ γ t , ˆ S r,t = S r \ ˆ γ t . If t < T ∧ τ r , there is a unique r ( t ) = r ( t, γ t ) ∈ (0 , r ] such that there is aconformal transformation ¯ h t : A r \ η t → A r ( t ) , with ¯ h t ( C ) = C . The transformation ¯ h t is unique up to a rotation. This transfor-mation can be raised to the covering space S r to give a conformal transformation h t : ˆ S r,t → S r ( t ) with h t ( ±∞ ) = ±∞ . This transformation is unique up to a real translation, andwe specify it uniquely by requiring h t ( U t ) = U t . We define φ t by h t = φ t ◦ g t . Note that φ t is the unique conformal transformation of g t ( S r,t ) onto S r ( t ) with φ t ( ±∞ ) = ±∞ and φ t ( U t ) = U t . Although r ( t ) depends on the curve γ , the nextlemma shows that its derivative at 0 is independent of γ assuming γ has the capacityparametrization. Lemma 3.12. If γ is a curve with hcap[ γ t ] = at , then ˙ r (0) = − a/ − /κ. Proof.
We will consider excursion measure defined by E D ( V , V ) = 12 π Z V Z V H D ( z, w ) | dz | | dw | . This definition assumes V , V are nice boundaries, but this is a conformal invariant(see [10, Chapter 5]) and hence is defined for rough boundaries as well. In thisnormalization, E r := E A r ( C , C r ) = 1 /r . Consider D t = A r \ η t where η = ψ ◦ γ . We only need to consider small t for which η is a simple curve in A r . Let E ( t ) = E D t ( C r , C ∪ η t ). By definition of r ( t ) and conformal invariance of excursionmeasure, E ( t ) = 1 /r ( t ) . Therefore, by the chain rule(37) ˙ E (0) = ˙ r (0) r . Suppose r > t is sufficiently small so that D ⊂ D t . Then using the strongMarkov property, E A r ( C r , C ) − E D ( C r , D t ) = E D ∩ A r ( C r , C ) E [ q ( B τ t )] = 1 r − E (cid:20) − log | B τ t | r (cid:21) . Here B is a Brownian motion started uniformly on C s , τ t is the first time that itleaves D t and q ( z ) denotes the probability that a Brownian motion starting at z hits C r before C , q ( z ) = − log | z | r . Therefore, ˙ E (0) = 1 r rr − ∂ t E [log | B ρ t ∧ σ r | ] | t =0 . where ρ t is the first time to leave D t and σ r is the first time to hit C r . We claimthat(38) ∂ t E [log | B ρ t ∧ σ r | ] | t =0 = r − r ∂ t E [log | B ρ t | ] | t =0 . To see this, we first note that the probability starting at C of hitting C r before C is 1 /r . Also, given ρ t < σ r , the probability of hitting C r before C is O ( d t /r )where d t = diam( γ t ) = o (1). Also, since we start with the uniform distribution on C , the distribution of σ r given that σ r < σ is also uniform. Therefore, E [log | B ρ t | ; σ r < ρ t ] = 1 r E [log | B ρ t | ] [1 + O ( d t )] . and hence ∂ t E [log | B ρ t | ; σ r < ρ t ] | t =0 = 1 r ∂ t E [log | B ρ t | ] | t =0 . from which (38) follows. Note that the right-hand side of (38) is the same if westart the Brownian motion at the origin.By comparison with (37), we see that ˙ r (0) is independent of r (0), and we cancompute ˙ r (0) by letting r ↓
0. In this case, we get the comparison of the chordalLoewner equation to the radial Loewner equation. (cid:3)
LE IN MULTIPLY CONNECTED DOMAINS 31
We define σ s = inf { t : r ( t ) = s } . Let γ ∗ be γ with the “annulus parametrization” γ ∗ ( s ) = γ ( σ s ) , ≤ s ≤ r, and let U ∗ s = U r ( s ) , h ∗ s = h σ s . The direction of “time” is reversed so one must be careful with minus signs. ♣ In the annulus parametrization, the radius takes the place of time. However, the directionof “time” is reversed, so one must take some care with minus signs.
We will just state the annulus Loewner equation (see, e.g., [1, 7]). It can alsobe described in terms of excursion reflection Brownian motion (this helps motivatethe formulas), see [3, 13]. We review the facts here. Let H S r ( z, x ) = H S r ( z − x )denote the complexification of the Poisson kernel in S r which recall by (4) is givenby H S r ( z ) = − π r coth (cid:16) πz r (cid:17) , and satisfies Im H ( z ) = H S r ( z, , H S r ( z ) = − z + O ( | z | ) , z → , and if x ∈ R ,Re H S r ( x ) = − π r coth (cid:16) πx r (cid:17) , Re H S r ( x + ir ) = − π r tanh (cid:16) πx r (cid:17) . There exists a unique holomorphic function with period 2 π H r : S r → H r , such that H r ( z ) = − z + o (1) , z → , and such that the induced map ¯ H r ( e iz ) = H r ( z )is a conformal transformation of A r onto a domain of the form H \ L for somehorizontal line segment L . One can find this using excursion reflected Brownianmotion (ERBM) as we now sketch. The imaginary part H r = Im H r will be thePoisson kernel for ERBM in the annulus. We can write(39) H r ( z ) = Im( z )2 r + H A r ( e iz ,
1) = Im( z )2 r − π r X k ∈ Z Im coth (cid:16) πz r (cid:17) . In this formula, the infinite sum represents the contribution to the ERBM Poissonkernel by paths that do not hit the “hole” D \ A r . The first term gives the contri-bution of paths that hit the hole first. The probability of hitting the hole beforehitting C is Im( z ) /r . Given that it hits the hole, the distribution of the first visitto C is uniform on the circle and hence the value of the kernel is 1 / H D (0 ,
1) = 1 / One can check that the sum in (39) absolutely convergent. However, the realparts are not absolutely convergent so we must take a little care in the definitionof H r . We write H r ( z ) = z r − π r coth (cid:16) z π r (cid:17) − π r ∞ X k =1 (cid:20) coth (cid:18) ( z + 2 kπ ) π r (cid:19) + coth (cid:18) ( z − kπ ) π r (cid:19)(cid:21) = z r − π r P P X k coth (cid:18) ( z + 2 kπ ) π r (cid:19) , where we write P P X k f ( k ) = lim N →∞ N X k = − N f ( k ) . Lemma 3.13. As z → , (40) H r ( z ) = − z + z (cid:18) r − Γ( r ) + 112 (cid:19) + O ( | z | ) , where Γ( r ) is as defined in (20) .Proof. We use the first expression for the definition of H r . Note that as z → z = 1 z + z O ( | z | ) , and hence π r coth (cid:16) z π r (cid:17) = π r (cid:20) rzπ + zπ r + O ( | z | ) (cid:21) = 1 z + π z r + O ( | z | )Also the derivative at z = 0 of − π r ∞ X k =1 (cid:20) coth (cid:18) ( z + 2 kπ ) π r (cid:19) + coth (cid:18) ( z − kπ ) π r (cid:19)(cid:21) is − δ ( r ) . (cid:3) Note that H r ( z + ir ) = z + ir r − π r P P X k tanh (cid:18) ( z + 2 kπ ) π r (cid:19) = − H I ( r, x )2 + i , where H I is as defined in Section 3.2.The chordal equation (16) can be written as ∂ t g t ( z ) = − a H H ( g t ( z ) − U t ) . The annulus Loewner equation is similar, ∂ t h t ( z ) = 2 ˙ r ( t ) H r ( t ) ( h t ( z ) − U t ) , or equivalently,(41) ∂ r h ∗ r ( z ) = 2 H r ( h ∗ r ( z ) − U ∗ r ) . LE IN MULTIPLY CONNECTED DOMAINS 33
An important observation is that if r (0) = r , then for small t , the functions g t , h t ,and h ∗ r − at are very close near the origin. For future reference, we also note that(42) ∂ s log( h ∗ s ) ′ ( x + ir ) | s = r = 2 H ′ r ( x + ir ) = − H ′ I ( r, x ) . ♣ There may appear to be some arbitrariness in the choice of the real translation for thecomplex kernel H H ( g t ( z ) − U t ) . It turns out that this choice is not so important. We will write d [ h ∗ r ( z ) − U ∗ r ] = 2 H r ( h ∗ r ( z ) − U ∗ r ) − dU ∗ r . If we had chosen a different real translation of H r , it would cancel here when we took thedifference. ♣ We have written the annulus equation in the covering space S r . We would also considerthe function given by f s ( e iw ) = e ih s ( w ) , ≤ s ≤ r. There is a curve η : (0 , r ) → A r with η (0+) = 1 such that f s is a conformal transformation of A r \ γ s onto A r − s . Such a transformation is defined up to a rotation, but specifying continuityand f s ( η ( r − s )) = U ∗ s determines the rotation. We will need to compare the chordal and annulus equations at time t = 0. Recallthat φ t is defined by h t ( z ) = φ t ( g t ( z )) , and that φ t ( U t ) = U t = g t ( γ ( t )). Although g t is not smooth at γ ( t ), it is notdifficult to show that φ t is analytic in a neighborhood of U t and we can give thederivatives. We summarize the facts we need in this lemma whose simple prove weomit. Lemma 3.14.
Suppose K j,t ( z ) , j = 1 , , t ∈ [0 , ǫ ] are analytic functions in a punc-tured neighborhood of the origin and are continuous in t . Suppose U t is a continuousfunction with U = 0 and g t , h t satisfy ∂ t g t ( z ) = K ,t ( g t ( z ) − U t ) , ∂ t h t ( z ) = K ,t ( h t ( z ) − U t ) , with g ( z ) = h ( z ) . Suppose that for all t , K ,t − K ,t is analytic in the (unpunc-tured) neighborhood. If φ t is defined by h t ( z ) = φ t ( g t ( z )) , then (43) ˙ φ ( z ) = [ K , − K , ]( z ) , ˙ φ ′ ( z ) = [ K , − K , ] ′ ( z ) . We now return to the locally chordal
SLE κ from 0 to z = x + ir in S r . Giventhe path γ t , the process is moving infinitesimally like SLE κ in ˆ S r,t from γ ( t ) to z . By conformal invariance we can also view it in g t ( ˆ S r,t ) from U t to g t ( z ) or in h t ( ˆ S r,t ) = S r ( t ) from U t to h t ( z ). Using the last perspective and (31) and (32), wesee that dU t = b L ( r ( t ) , R t ) dt − dW t , where R t = Re[ h t ( z )] − U t and W t is a standard Brownian motion. We choose atime parametrization so that the radius evolves linearly. If U ∗ t = U σ ( t ) as above, dU ∗ t = bκ L ( r − t, R ∗ t ) dt − √ κ dB t . Using (42), we see that if f t = h ∗ r − t , ∂ t [Re f t ( z )] = H I ( r − t, R ∗ t ) , and hence(44) dR ∗ t = [ H I ( r − t, R ∗ t ) − bκ L ( r − t, R ∗ t )] dt + √ κ dB t . We have written locally chordal
SLE κ in the annulus as a one-dimensional SDEstopped at a finite time r . The next lemma shows that the process leaves S r at z .The equivalent statement is the following. Lemma 3.15. If X t satisfies dX t = [ H I ( r − t, X t ) − bκ L ( r − t, X t )] dt + √ κ dB t , ≤ t < r, then with probability one X r − = 0 . ♣ This lemma should not be surprising. If we considered chordal
SLE κ from to x + ir in S r we know that (for κ ≤ ) the path leaves the domain at x + ir . This lemma stays thatthe same thing for locally chordal SLE κ . Since for r near zero, locally chordal and chordal SLE κ are almost the same, the lemma has to be true. One should expect κ ≤ to come intothe proof, and this is the case. Proof.
We discuss the most delicate case, κ = 4 for which bκ = 1; if κ <
4, then bκ > dX t = [ H I ( r − t, X t ) − L ( r − t, X t )] dt + 2 dB t . If Y s = X r − e − s , then Y s satisfies dY s = m ( s, Y s ) ds + 2 e − s/ dW s , where m ( s, y ) = e − s (cid:2) H I ( e − s , y ) − L ( e − s , y ) (cid:3) , and W s is a standard Brownian motion. It suffices to show that for every ǫ > | Y s | ≤ ǫ for all s sufficiently large. By symmetry it suffices toshow that that lim sup Y s ≤
0. Let Z s = Z s e − r/ dW r , and note that with probability one Z ∞ exists and is finite.Using Lemma 3.2, we can see that there exists s ǫ such that m ( s, y ) ≤ s ≥ s ǫ , y ≥ ǫ/
2. Therefore, if Y s ≥ ǫ and s ≥ s ǫ , Y r ≤ ǫ + max t ≥ s ǫ | Z t − Z s ǫ | . Therefore, it suffices to show that with probability one lim inf Y n ≤
0. In otherwords, for every ǫ > , s < ∞ , y >
0, the probability that the process reaches ǫ given Y s = y equals one.Although the drift m ( s, y ) is negative, the absolute value is very small at y slightly larger than an integer multiple of 2 π . However, we also know from Lemma3.2 that for all y , m ( s, y ) ≤ − c e − s . Given this, we can see that if we start near 2 πk ,there is at least a positive probability that there will exist s with Y s ≤ πk − c e − s .Given this, there is a positive probability that the process will never return to { y ≥ πk − ( c / e − s } and since the drift is negative, this will imply that it willget near 2 π ( k − LE IN MULTIPLY CONNECTED DOMAINS 35 are near 2 πk at a larger time s ′ we can find s ′′ > s ′ for which Y s ′′ ≤ πk − c e − s ′′ .Eventually we will succeed and get to 2 π ( k − (cid:3) Definition of µ D ( z, w )4.1. Definition of boundary
SLE κ for κ ≤ . We fix κ ∈ (0 , SLE κ as proposed in [14]. It is a (positive) measure µ D ( z, w ) on simple curves γ in a domain D connecting distinct ∂D -analytic bound-ary points z and w . If D is simply connected, then the definition is the same asthat of chordal SLE κ . We writeΨ D ( z, w ) = k µ D ( z, w ) k for the total mass of the measure. We conjecture that Ψ D ( z, w ) < ∞ for all D, z, w .In the case of simply connected domains, we know this is true, and in this paperwe will show it for 1-connected domains for κ ≤
4. From the construction it willfollow that Ψ D ( z, w ) < ∞ for all domains if κ ≤ / c ≤ D ⊂ D is a subdomain of D that agrees with D in neighborhoods of z and w . We let µ D ( z, w ; D ) be µ D restricted to curves γ ⊂ D . LetΨ D ( z, w ; D ) = k µ D ( z, w ; D ) k . We will show that µ D ( z, w ; D ) < ∞ for all such simply connected D for κ ≤ µ D ( z, w ; D ) is defined to be the probability measure obtained bynormalization µ D ( z, w ; D ) = µ D ( z, w ; D )Ψ D ( z, w ; D ) . If Ψ D ( z, w ) < ∞ , we write µ ( z, w ) for the probability measure. ♣ What we call boundary
SLE should really be called boundary/boundary
SLE , butthis terminology is a bit cumbersome. In later subsections, we also discuss boundary/bulk,bulk/boundary, and bulk/bulk cases.
In this definition and later on we use the convention as described below equa-tion (3) that if formulas are written with derivatives, then sufficient smoothness isassumed.
Definition If κ ≤ b, c are as in (1), boundary SLE κ is the unique familyof measures (modulo reparametrization) { µ D ( z, w ) } , where D ⊂ C and z, w aredistinct ∂D -analytic points, satisfying the following. • For each
D, z, w , µ D ( z, w ) is a positive measure on curves γ : [0 , t γ ] → D with γ (0) = z, γ ( t γ ) = w, γ ⊂ D . The total mass is denoted byΨ D ( z, w ) = k µ D ( z, w ) k . The normalization is chosen so that Ψ H (0 ,
1) = 1. • Conformal covariance If f : D → f ( D ) is a conformal transformation,then(45) f ◦ µ D ( z, w ) = | f ′ ( z ) | b | f ′ ( w ) | b µ f ( D ) ( z, w ) . • It follows from (45) that the probability measures are conformally invariant, f ◦ µ D ( z, w ; D ) = µ f ( D ) ( f ( z ) , f ( w ); f ( D )) , and if Ψ D ( z, w ) < ∞ ,(46) f ◦ µ D ( z, w ) = µ f ( D ) ( z, w ) . In particular, µ D ( z, w ; D ) (resp., µ D ( z, w )) can be defined for nonan-alytic boundary points provided that there is a conformal transforma-tion f : D → f ( D ) such that f ( z ) , f ( w ) are ∂f ( D )-analytic (resp., withΨ f ( D ) ( f ( z ) , f ( w )) < ∞ ). • Domain Markov property . If Ψ D ( z, w ) < ∞ , then for the probabilitymeasure µ D ( z, w ), the conditional probability measure of the remainder ofa curve γ given an initial segment γ t , is that of µ D \ γ t ( γ ( t ) , w ). If D ⊂ D issimply connected, for the probability measure µ D ( z, w ; D ), the conditionalprobability measure of the remainder of a curve γ given an initial segment γ t , is that of µ D \ γ t ( γ ( t ) , w ; D \ γ t ). • Boundary perturbation . Suppose D ′ ⊂ D are domains that agree inneighborhoods of ∂D ′ -analytic boundary points z, w . Then µ D ′ ( z, w ) isabsolutely continuous with respect to µ D ( z, w ) with Radon-Nikodym de-rivative Y = Y D,D ′ ,z,w given by(47) Y ( γ ) = dµ D ′ ( z, w ) dµ D ( z, w ) ( γ ) = 1 { γ ⊂ D ′ } exp n c m D ( γ, D \ D ′ ) o . We will now construct the measure and in the process show uniqueness. Forsimply connected domains, we set Ψ D ( z, w ) = H ∂D ( z, w ) b and µ D ( z, w ) to be theconformal image of Ψ H (0 , ∞ ) under a conformal transformation. The discussionin Section 3.6 shows that this is the unique family of measures that satisfy theconditions above for simply connected D . Definition
Suppose D is a domain and z, w are distinct ∂D -analytic boundarypoints. Let D be a simply connected subdomain of D that agrees with D inneighborhoods of z, w . Then ˆ µ D ( z, w ; D ) is the measure absolutely continuouswith respect to µ D ( z, w ) with Radon-Nikodym derivative(48) d ˆ µ D ( z, w ; D ) dµ D ( z, w ) ( γ ) = 1 { γ ⊂ D } exp n − c m D ( γ, D \ D ) o . ♣ A minus sign appears on the right-hand side above. This is because we are writing thederivative of the measure on the larger domain with respect to that on the smaller domain.
The next proposition establishes a necessary consistency condition for the mea-sures ˆ µ D ( z, w ; D j ) in order to define µ D ( z, w ). Proposition 4.1.
Suppose D is a domain and z, w are distinct ∂D -analytic bound-ary points. Let D , D be simply connected subdomains of D that agree with D inneighborhoods of z, w . For j = 1 , , let ν j be ˆ µ D ( z, w ; D j ) restricted to curves γ with γ ⊂ D ∩ D . Then ν = ν . LE IN MULTIPLY CONNECTED DOMAINS 37
Proof.
Suppose γ ⊂ D ∩ D . Then there exists simply connected ˆ D ⊂ D ∩ D that agrees locally with D near z, w such that γ ⊂ ˆ D . Hence it suffices to showthat for every simply connected domain ˆ D , ν and ν , restricted to curves in ˆ D ,agree. Suppose γ ⊂ ˆ D . Since D j , ˆ D are simply connected, dµ D j ( z, w ) dµ ˆ D ( z, w ) ( γ ) = exp n − c m D j ( γ, D j \ ˆ D ) o . Combining this with (48), we get d ˆ µ D ( z, w ; D j ) dµ ˆ D ( z, w ) ( γ ) = exp n − c m D ( γ, D \ ˆ D ) o . Here we use the fact that the loops in D that intersect γ and D \ ˆ D can be partitionedinto two sets: those that intersect D \ D and those that are contained in D . (cid:3) Given Proposition 4.1 we can make the following definition.
Definition
Suppose D is a domain and z, w are distinct ∂D -analytic boundarypoints. Then µ D ( z, w ) is the measure on simple paths (modulo parametrization)such that for each simply connected D ⊂ D , µ D ( z, w ) restricted to curves γ ⊂ D is ˆ µ D ( z, w ; D ).In other words, µ D ( z, w ; D ) = ˆ µ D ( z, w ; D ) for simply connected D . It followsimmediately from the definition that the family of measures { µ D ( z, w ) } satisfies(47). Suppose D is a domain and z, w are distinct ∂D -analytic points and D is a simply connected domain as above. Suppose f : D → f ( D ) is a conformaltransformation. Then f : D → f ( D ) is also a conformal transformation, andhence f ◦ µ D ( z, w ) = | f ′ ( z ) | b | f ′ ( w ) | b µ f ( D ) ( f ( z ) , f ( w )) . Conformal invariance of the loop measure then implies that f ◦ µ D ( z, w ; D ) = | f ′ ( z ) | b | f ′ ( w ) | b µ f ( D ) ( z, w ; f ( D )) . Since this is true for every simply connected D , the family { µ D ( z, w ) } satisfies(45).In this paper, we will show the following. (While we prove it in this paper, wecould also derive this from [25].) Proposition 4.2. If D is a conformal annulus, then Ψ D ( z, w ) < ∞ and the family { µ D ( z, w ) } restricted to conformal annuli satisfies the domain Markov property. When considering the measure µ D ( z, w ) for multiply connected domains, thereare two cases. • The chordal case: z, w in the same component of ∂D . Then there existssimply connected ˆ D such that D ⊂ ˆ D . • The crossing case: z, w in different components of ∂D . Then there exists1-connected ˆ D such that D ⊂ ˆ D . Proposition 4.3.
Suppose D is a domain and z, w are distinct ∂D -analyticpoints. • If κ ≤ / , then Ψ D ( z, w ) < ∞ . • If / < κ ≤ , then for every simply connected D ⊂ D that agrees with D near z, w , Ψ D ( z, w ; D ) < ∞ . Proof. If κ ≤ /
3, we can consider D as a subdomain of a simply connected or 1-connected domain ˆ D and since c ≤
0, (47) implies that Ψ D ( z, w ) ≤ Ψ ˆ D ( z, w ) < ∞ .If 8 / < κ ≤
4, then c >
0, and (48) implies that Ψ D ( z, w ; D ) ≤ Ψ D ( z, w ) < ∞ . (cid:3) Proposition 4.4.
The family { µ D ( z, w ) } satisfies the domain Markov property.Proof. Without loss of generality we may assume that D is a subdomain of H whoseboundary includes R and z = 0. Let D be a simply connected domain as abovefor which we know Ψ D ( z, w ; D ) < ∞ and let γ t be an initial segment. To be moreprecise, let t be a finite stopping time for chordal SLE κ in D . Let F t be thecorresponding σ -algebra generated by γ t . For γ ⊂ D , let Y ( γ ) = µ D ( z, w ; D ) µ D ( z, w ) ( γ ) = exp n c m D ( γ, D \ D ) o . Let P , E denote probability and expectation with respect to the probability measure µ D ( z, w ). Then, Ψ D ( z, w ; D ) = Ψ D ( z, w ) E [ Y ] . By the domain Markov property for
SLE κ is simply connected domains, E [ Y | F t ] = exp n c m D ( γ t , D \ D ) o E ∗ t [ Y ] , where E ∗ t denotes expectation with respect to µ D \ γ t ( γ ( t ) , w ).We will do the chordal case comparing to simple connected domains. The cross-ing case is similar using conformal annuli. Suppose z, w are in the same componentof ∂D . Without loss of generality, we may assume that D is a subdomain of H and z, w ∈ R . We know that dµ D ( z, w ) dµ H ( z, w ) ( γ ) = 1 { γ ∈ D } exp n c m H ( γ, H \ D ) o . Let P , E denote probabilities and expectations with respect to the measure µ H ( z, w ).Let Y t = 1 { γ t ⊂ D } exp n c m H ( γ t , H \ D ) o , Y = Y ∞ . Suppose we are given an initial segment γ t and let H t = H \ γ t . Here t can bea stopping time and we assume that t < T = inf { s > γ ( s ) ∈ R } = inf { s > γ ( s ) = w } . (The equality is true with P probability one.) Let g t denote thecorresponding map and let F = F t denote the σ -algebra generated by t . By thedomain Markov property of SLE κ in simply connected domains, E [ Y | F ] = Y t E ∗ t h exp n c m H t ( γ, H t \ D ) oi , where E ∗ t denotes expectations with respect to µ H t ( γ ( t ) , w ). More generally if E isan event depending on the path γ \ γ t , E [ Y E | F ] = Y t E ∗ t h E exp n c m H t ( γ, H t \ D ) oi , If Ψ D ( z, w ) < ∞ , the proof for µ D ( z, w ) is similar and we omit it. (cid:3) Proposition 4.5. If z, w are ∂D -analytic, then µ D ( w, z ) is the same as the reversalof µ D ( z, w ) . LE IN MULTIPLY CONNECTED DOMAINS 39
Proof.
In the case of simply connected domains, this was proved by Zhan [24].Given this, the general case follows. (cid:3)
We end this section with a number of remarks. • In our definition we have started with the parameter κ and defined thequantities b, c in terms of κ . We could have made b, c free parameters, butthen we would find out that there was only a one-dimensional family ofpairs ( b, c ) for which we could define such measures. To establish this fact,we would use Schramm’s argument and κ (as a function of b or c ) wouldbe introduced. • Implicit in the domain Markov property is the assumption that the theinitial segment may be chosen using a stopping time. This makes it acondition on curves modulo reparametrization. Perhaps this should becalled the strong domain Markov property. • It is also useful to have the measures µ D ( x, ∞ ) where D ⊂ H with H \ D bounded and dist( x, H \ D ) >
0. To get this we find a conformal transfor-mation f : D ′ → D with f ( z ) = 0 , f ( w ) = ∞ and use the conventions about derivatives as inSection 3.1. Under this convention, we see that Ψ H (0 , ∞ ) = 1. If D ⊂ H is simply connected with H \ D bounded and dist(0 , H \ D ) >
0, thenΨ D (0 , ∞ ) = Φ ′ D (0) b where Φ D : D → H is a conformal transformationwith Φ D ( ∞ ) = ∞ , Φ ′ D ( ∞ ) = 1.4.2. Definition of boundary/bulk and bulk/bulk
SLE κ for κ ≤ . Theboundary
SLE κ is a measure on curves connecting two boundary points in a do-main D . We extend this definition to allow one boundary point and one interiorpoint (the radial or reverse radial case) or two interior points (the bulk case). In allthe cases we will write µ D ( z, w ) for the measure, Ψ D ( z, w ) for the total mass, and ifΨ D ( z, w ) < ∞ µ D ( z, w ) for the corresponding probability measure. The definitionwill be the same as the first definition in Section 4.1 except that (45) is replacedwith the following more general formula. Note that this definition subsumes theprevious one. • Conformal covariance If f : D → f ( D ) is a conformal transformation, z, w are D -analytic, and f ( z ) , f ( w ) are f ( D )-analytic, then(49) f ◦ µ D ( z, w ) = | f ′ ( z ) | b z | f ′ ( w ) | b w µ f ( D ) ( z, w ) , where b ζ = b if ζ is a boundary point and b ζ = ˜ b if ζ is an interior point. ♣ We are writing µ D ( z, w ) for all the cases in order not to add more notation. It isimportant to remember that the definitions of these measures are different (although related,of course) depending on whether z, w are boundary or interior points. If D is simply connected, z is ∂D -analytic and w ∈ D , then we define µ D ( z, w )by µ D ( z, w ) = Ψ D ( z, w ) µ D ( z, w ) , where µ D ( z, w ) is radial SLE κ as in Section 3.3. The partition function Ψ D ( z, w )is determined up to a multiplicative constant by the rule (49), and we choose the constant so that Ψ D (1 ,
0) = 1. Using the relationship in Section 3.8, onecan check that this satisfies the necessary conditions. In particular, the boundaryperturbation rule (47) holds for simply connected domains.It was essentially shown in [25], and we will reprove it here, that radial
SLE κ canbe given as a limit of boundadry/boundary SLE κ in the annulus. The followingtheorem makes a more precise estimate. Theorem 4.6.
There exists c < ∞ , q > such that the following holds. Let t > and let γ t denote an initial segment of a path in D starting at such that if g : D \ γ t → D is a conformal transformation with g (0) = 0 , g ′ (0) > , then g ′ (0) = e t . Suppose that r ≥ t + 2 , ≤ θ < π , and let µ = µ D (1 , , µ = µ A r (1 , e − r + iθ ) , bothconsidered as probability measures on initial segments γ t . Let Y = dµ /dµ . Then (50) | Y ( γ t ) − | ≤ c e ( t − r ) q . Moreover, there exists c ∈ (0 , ∞ ) such that (51) Ψ(1 , e − r + ix ) = c e ( b − ˜ b ) r r c / [1 + O ( e − qr )] . We will write µ A r (1 , e − r + ix ) = c e ( b − ˜ b ) r r c / µ D (1 ,
0) [1 + O t ( e − qr )] , as shorthand for (50) and (51). ♣ We can see the interior scaling exponent as coming from a computation from the annuluspartition function. Suppose D is a bounded domain, ∈ D and w ∈ ∂D is D -analytic.Suppose that ǫ is small and | z | = ǫ . Let D ǫ denote the conformal annulus obtained byremoving the closed disk of radius ǫ . Then by analysis of the annulus partition function whichis a boundary/boundary quantity, we see as ǫ → , Ψ D ǫ (1 , z ) ∼ c ǫ ˜ b − b [log(1 /ǫ )] c / , and hence we can define Ψ D (1 , (up to an arbitrary multiplicative constant) by Ψ D (1 , ∼ ǫ b − ˜ b [log(1 /ǫ )] − c / Ψ D ǫ (1 , ǫ ) . If f : D → f ( D ) is a conformal transformation with f (0) = 0 , then f ( D ǫ ) is approximatelythe disk of radius f ′ (0) ǫ , and Ψ D ǫ (1 , z ) ∼ | f ′ (1) | b | f ′ ( z ) | b Ψ D ǫf ′ (0) ( f (1) , f ( z )) ∼ | f ′ (1) | b | f ′ (0) | b Ψ D ǫf ′ (0) ( f (1) , f ( z )) Therefore, if u = | f ′ (0) | , Ψ D (1 , ∼ ǫ b − ˜ b [log(1 /ǫ )] − c / Ψ D ǫ (1 , z ) ∼ | f ′ (1) | b u ˜ b ( uǫ ) b − ˜ b [log(1 /ǫ )] − c / Ψ f ( D ) uǫ ( f (1) , f ( uz )) Note that the logarithmic term which includes the central charge does not contribute to thescaling exponent.
We now define boundary/bulk and bulk/boundary
SLE . The consistency of thisdefinition follows from the fact that (47) holds for simply connected domains.
Definition If z ∈ D and w is a ∂D -analytic boundary point, then µ D ( w, z ) and µ D ( z, w ) are defined as follows. LE IN MULTIPLY CONNECTED DOMAINS 41 • If D is simply connected, µ D ( w, z ) = | f ′ ( w ) | − b | f ′ ( z ) | − ˜ b f ◦ µ D (1 , , where f : D −→ D is the conformal transformation with f (1) = w, f (0) = z . • If D ⊂ D where D is simply connected and agrees with D near z and w ,then dµ D ( w, z ) dµ D ( w, z ) ( γ ) = 1 { γ ⊂ D } exp n c m D ( γ, D \ D ) o . • µ D ( z, w ) is defined to be the measure obtained from µ D ( w, z ) by reversingthe paths.We can define bulk/bulk SLE κ similarly. There is technical issue if D is all of C . Let us define D to be regular if with probability one a Brownian motion exitsthe domain D . Definition If z, w are distinct points of a regular domain D , then µ D ( z, w ) isdefined by µ D ( z, w ) = c − lim r →∞ e b − b ) r r c / µ D r ( z + e − r , w + e − r ) , where D r = { ζ ∈ D : | ζ − z | > e − r , | ζ − w | > e − r } . We could also have defined µ D ( z, w ) = c − lim r →∞ e b − b ) r r c / µ D r ( z + e − r + iθ , w + e − r + iθ ′ ) , for any θ, θ ′ . Alternatively, we could define µ D ( z, w ) = c ′ lim r →∞ e (˜ b − b ) r µ D r,z ( z + e − r + iθ , w + e − r + iθ ′ ) , where D r,z = { ζ ∈ D : | ζ − z | > e − r } . Our choice of definition has the advantage that it follows immediately that µ D ( w, z )is the reversal of µ D ( z, w ). If we want to let D = C , we have to renormalize. Proposition 4.7. If z, w ∈ D , then There exists Ψ( z, w ) ∈ (0 , ∞ ) such that Ψ D − r ( z, w ) = Ψ( z, w ) r − c / [1 + O ( r − )] . Proof.
This essentially follows from Proposition 3.11. (cid:3)
Using this as a guide, we define µ ( z, w ) = c ′ lim r →∞ r c / µ A − r ( z, w ) . This satisfies the conformal covariance rule f ◦ µ ( z, w ) = | f ′ ( z ) | ˜ b | f ′ ( w ) | ˜ b µ ( f ( w ) , f ( w )) , where f is a linear fractional transformation (conformal transformation of the Rie-mann sphere). Conformal covariance implies that there exists c ′′ ∈ (0 , ∞ ) such thatfor all z, w , Ψ( z, w ) = c ′′ | z − w | − b . The probability measure µ ( z, w ), which is invariant under linear fractional trans-formations, is called whole plane SLE κ . While we have defined µ ( z, w ) as a limit,we could also imagine being able to define it directly. In this case, we get µ D ( z, w )by a (normalized) boundary perturbation rule. Proposition 4.8. If D is a domain and z, w ∈ D are distinct, then dµ D ( z, w ) dµ ( z, w ) ( γ ) = 1 { γ ⊂ D } exp n − c γ, ∂D ) o where Λ( γ, ∂D ) is as defined in Proposition 3.11.Proof. For r sufficiently large so that γ ⊂ D − r , dµ D ( z, w ) dµ A − r ( z, w ) ( γ ) = exp n c m A − r ( γ, A − r \ D ) o . Proposition 3.11 implies that as r → ∞ , m A − r ( γ, A − r \ D ) = log r − Λ( γ, ∂D ) + o (1) . Therefore, dµ D ( z, w ) r c / dµ A − r ( z, w ) ( γ ) ∼ exp n − c γ, ∂D ) o , r → ∞ . (cid:3) ♣ While it might seem natural to define µ ( z, w ) using whole plane SLE and then theproposition to define µ D ( z, w ) , there is a disadvantage in this approach. The reason is thatit is not so easy to prove that µ D ( z, w ) satisfies the conformal covariance relation for con-formal transformations of D since the quantity Λ( γ, ∂D ) is not conformally invariant undertransformations of D . Example If κ = 2, then Ψ D ( z, w ) is proportional to the usual Green’s functionfor Brownian motion with Dirichlet boundary conditions. For this, it is well knownthat Ψ A − r (0 , ∼ r, which agrees with the formula since c = −
2. Also, ˜ b = 0 which implies thatΨ D ( z, w ) is a conformal invariant . This is well known for the Green’s function.4.3. Multiple paths.
Extending the definition of
SLE κ to multiple is straight-forward as in [8]. Suppose z = ( z , . . . , z k ) , w = ( w , . . . , w k ) are distinct analyticpoints in a domain D . The points can be bulk or boundary points. The measure µ D ( z , w ) is defined by giving its Radon-Nikodym derivative Y with respect to theproduct measure µ D ( z , w ) × · · · × µ D ( z k , w k ) . Let ¯ γ = ( γ , . . . , γ k ) be a k -tuple of paths (modulo reparametrization) in D where γ j goes from z j to w j . Then(52) Y = 1 { γ j ∩ γ l = ∅ , j = l } exp c k X j =2 m D ( γ j , γ ∪ · · · ∪ γ j − ) . ♣ One can consider the measure on multiple paths in the context of the λ -SAW. On thediscrete level, the measure on a k -tuple of paths ¯ ω = ( ω , . . . , ω k ) is exp n − β ( | ω | + · · · + | ω k | ) + λ m RW ( ω ∪ · · · ∪ ω k , D, n ) o . The exponential factor on the right hand side of (52) compensates for overcounting of loopsthat intersect ¯ γ . LE IN MULTIPLY CONNECTED DOMAINS 43 Crossing
SLE κ in an annulus In this section we study the measure µ A r (1 , e − r + iθ ) which is a measure on simplepaths (modulo reparametrization) η from 1 to e − r + iθ in A r . Let us recall thedefinition. Suppose D ′ is a simply connected subdomain of A r that agrees with A r in neighborhoods of 1 and w = e − r + iθ . Then if η is a curve in D ′ connecting 1 and w , dµ A r (1 , w ) dµ D ′ (1 , w ) ( η ) = exp n − c m A r ( η, A r \ D ′ ) o . We can write(53) m A r ( η, A r \ D ′ ) = ˆ m A r ( η, A r \ D ′ ) + m ∗ ( r ) , where m ∗ ( r ) denotes the measure of the set of loops in A r of nonzero windingnumber and ˆ m A r ( η, A r \ D ′ ) is the measure of the set of loops of zero windingnumber that intersect both η and A r \ D ′ . Here we use the fact that every loop ofnonzero winding number intersects both η and A r \ D ′ . (This construction assumesthat there is a unique point on the Brownian loop that goes through the point η ( t ).For each curve η this is true up to a set of loops of measure zero. See the discussionafter Theorem 12 in [18].)Let γ be the continuous preimage under ψ of η with γ (0) = 0, and let D be thesimply connected domain containing γ such that ψ ( D ) = D ′ . Each loop ℓ ′ in A r has an infinite number of preimages under ψ . For each loop ℓ ′ in A r that intersects η , we choose a unique such preimage as follows. Consider the first time t such that η ( t ) ∈ ℓ ′ . We make ℓ ′ a rooted loop by choosing the root to be η ( t ). Then wechoose ℓ to be the (rooted) preimage of ℓ ′ that is rooted at γ ( t ). The definition of ℓ implies that if it is rooted at γ ( t ), then(54) ℓ ∩ ˜ γ t = ∅ , where, as before, ˜ γ t = [ k ∈ Z \{ } ( γ t + 2 πk ) . We will call a loop ℓ γ -good if it intersects γ and satisfies (54). Then ℓ ↔ ℓ ′ gives abijection between γ -good loops in S r and loops in A r of zero winding number thatintersect η .If r > x ∈ R , we define the measure ν S r (0 , x + ir ) by the relation dν S r (0 , x + ir ) dµ D (0 , x + ir ) ( γ ) = exp n − c m S r ( γ, S r \ D ; ∗ ) o , γ ⊂ D where m S r ( γ, S r \ D ; ∗ ) denotes the Brownian loop measure of γ -good loops in S r that intersect both γ and S r \ D . Recall that dµ S r (0 , x + ir ) dµ D (0 , x + ir ) ( γ ) = exp n − c m S r ( γ, S r \ D ) o , This leads to an alternative, equivalent definition of ν S r (0 , x + ir ). Note that ψ ◦ γ is a simple curve if and only if γ ∩ ˜ γ = ∅ . Definition
The measure ν S r (0 , x + ir ) is the measure absolutely continuous withrespect to µ S r (0 , x + ir ) with Radon-Nikodym derivative(55) dν S r (0 , x + ir ) dµ S r (0 , x + ir ) ( γ ) = 1 { γ ∩ ˜ γ = ∅} exp n c m S r ( γ ) o , where m S r ( γ ) is the measure of loops in S r that intersect γ but are not γ -good.We call this annulus SLE κ in S r from to x + ir .We can relate annulus SLE κ in S r to SLE κ in A r by conformal covariance. Wedefine ν A r (1 , x ) by ν A r (1 , x ) = | ψ ′ (0) | − b | ψ ′ ( x + ir ) | − b e − c m ∗ ( r ) / ψ ◦ ν S r (0 , x + ir )= e br e − c m ∗ ( r ) / ψ ◦ ν S r (0 , x + ir ) , (56)We think of this as annulus SLE κ from 1 to e − r + ix restricted to curves of a partic-ular winding number. The term e − c m ∗ ( r ) / is discussed in Proposition 3.9. Annulus SLE κ is obtained by summing over all winding numbers(57) µ A r (1 , e − r + iθ ) = X k ∈ Z ν A r (1 , θ + 2 πk ) . Main result.
We will show that the partition function for annulus
SLE on S r can be given in terms of a functional of locally chordal SLE κ . Recall the functions H I from Section 3.2, A from (14), and L from (32). Theorem 5.1. If ˜Ψ( r, x ) = k ν S r (0 , x + ri ) k , then ˜Ψ( r, x ) = V ( r, x ) Ψ S r (0 , x + ri ) . Here (58) V ( r, x ) = E x (cid:20) exp (cid:26) − b Z r A ( r − s, X s ) ds (cid:27)(cid:21) , where X t , ≤ t ≤ r satisifes (59) dX t = [ H I ( r − t, X t ) − bκ L ( r − t, X t )] dt + √ κ dB t , and B t is a standard Brownian motion. In particular, ˜Ψ( r, x ) is C in r , C in x and ˜Ψ( r, x ) ≤ Ψ S r (0 , x + ri ) . We used the functional in (58) as our definition, but as we show now, it is thesolution of a PDE. Let us define V (0 , x ) ≡ Proposition 5.2.
The function V ( r, x ) satisifes ≤ V ( r, x ) ≤ , is continuous on [0 , ∞ ) × ( − π, π ) and for r > satisfies the equation (60) ˙ V = − b A V + [ H I − bκ L ] V ′ + κ V ′′ , where dot refers to r -derivatives and primes refer to x -derivatives.Moreover, for fixed r , x V ( r, x ) is an odd function that is decreasing in | x | .Proof. For r >
0, the function H I , L are smooth and A ≥
0. Hence (60) followsfrom the Feynman-Kac formula, see, e.g, [5, Section 6.5] or [6, Section 5.7.b]. Com-bining (15) with Lemma 3.15, we see that V (0+ , x ) = 1 for | x | < π . For the lastassertion, we use Proposition 3.4 which states that A ( r, x ) is an increasing functionof | x | . It is not difficult to see that if 0 < x < x < ∞ , then we can couple process LE IN MULTIPLY CONNECTED DOMAINS 45 X t , X t on the same probability space, each satisfying (74) with X j = x j and suchthat | X t | ≤ | X t | for all t . In this coupling, we have Z r A ( r − s, X s ) ds ≤ Z r A ( r − s, X s ) ds. (cid:3) Radon-Nikodym derivative.
Similarly to the approach for simply con-nected domains as in Section 3.6, we will find an appropriate nonnegative localmartingale and use the Girsanov theorem to analyze the process weighted by thelocal martingale. Suppose (Ω , F , ˆ P ) is a probability space under which U t = − B t isa standard Brownian motion. Let g t be the solution to the Loewner equation (16)producing the random curve γ . Let γ t , ˜ γ t , ˆ γ t be as above and fix r, z = x + ir . Thefollowing proposition is the particular case of Section 3.6 for D = D r , w = x + ir .. Proposition 5.3. If J t = | g ′ t ( z ) | b H ∂g t ( S r \ γ t ) ( U t , g t ( z )) b exp n c m H ( γ t , H \ S r ) o , then J t is a local martingale for t < τ r . Moreover, if one uses Girsanov, then underthe weighted measure γ has the distribution of SLE κ from to x + ir . Let P , E denote expectations in the weighted measure under which γ has thedistribution of µ S r (0 , x + ir ).If ℓ is an (unrooted) loop in S r , let˜ s ( ℓ ) = min { t : ℓ ∩ ˜ γ t = ∅} ,s ( ℓ ) = min { t : ℓ ∩ γ t = ∅} . It is not hard to show, using the fact that two-dimensional Brownian motion doesnot hit points, that the loop measure of the set of loops with s ( ℓ ) = ˜ s ( ℓ ) < ∞ iszero. Let Λ t = Λ t ( γ t , r ) = 1 { T > t } exp { m t } , where m t = m t,r ( γ t ) denotes the measure of the set of loops in S r that satisfy˜ s ( ℓ ) < s ( ℓ ) ≤ t. Theorem 5.1 can be rephrased as follows.
Theorem 5.4. If γ has distribution µ S r (0 , x + ir ) , then (61) E h Λ c / τ r i = V ( r, x ) . We will prove (61) in a series of propositions. Recall the definition of A from(14). Let R t = Re[ h t ( z )] − U t , V t = V ( r ( t ) , R t ) ,Q t = Q S r \ γ t ( γ ( t ) , z ; S r \ ˆ γ t ) , K t = exp (cid:26) Z t ˙ r ( s ) A ( r ( s ) , R s ) ds (cid:27) . (62) N t = Λ c / t Q bt K abt , O t = K − abt V t ,M t = N t O t = Λ c / t Q bt V t . By conformal invariance, H ∂g t ( S r \ ˆ γ t ) ( U t , g t ( z )) = Q t H ∂g t ( S r \ γ t ) ( U t , g t ( z )) . Therefore, φ ′ t ( U t ) | φ ′ t ( g t ( z )) | H ∂h t ( S r \ ˆ γ t ) ( U ∗ t , h t ( z )) = Q t H ∂g t ( S r \ γ t ) ( U t , g t ( z )) , and hence φ ′ t ( U t ) | h ′ t ( z ) | H ∂h t ( S r \ ˆ γ t ) ( U ∗ t , h t ( z )) = | g ′ t ( z ) | Q t H ∂g t ( S r \ γ t ) ( U t , g t ( z )) . Therefore, we can write J t N t = C t ( z ) H ∂h t ( S r \ ˆ γ t ) ( U ∗ t , h t ( z )) b , where C t ( z ) = φ ′ t ( U t ) − b | h ′ t ( z ) | b exp n − c m H ( γ t , H \ S r ) o Λ c / t K abt , Important observations are that C t ( z ) is C in t and C t ( z ) = C t ( z + 2 π ). Lemma 5.5.
Suppose γ is a parametrized with hcap[ γ (0 , t ]] = at . Let ˜ γ t be asabove and Q t = Q S r \ γ t ( γ ( t ) , x + ir ; S r \ ˆ γ t ) . Then ∂ t Q t | t =0 = − a A ( r, x ) . Proof.
See (13). (cid:3)
Proposition 5.6. • N t is a local martingale with respect to P for t < T ∧ τ r . In particular, J t N t is a ˆ P -local martingale. • With respect to P ∗ , the curve γ at time t grows like SLE κ from γ ( t ) to z in ˜ S t,r .Proof. This is a particular case of Section 3.7. (cid:3)
Let P ∗ , E ∗ denote the probabilities and expectations obtained from P by weight-ing by the local martingale N t . This is the same as the measure obtained from ˜ P by weighting by J t N t . We have seen that this is locally chordal SLE κ and we canconsider the path in the annulus parametrization. Proposition 5.7.
Suppose V is as defined in (58) . Then M ∗ t = exp (cid:26) − b Z t A ( r − s, R ∗ s ) ds (cid:27) V ( r − t, R ∗ t ) , is a local martingale satisfying (63) dM ∗ t = √ κ V ′ ( r − t, R ∗ t ) V ( r − t, R ∗ t ) M ∗ t dB t . Moreover, if we weight by the local martingale using Girsanov theorem then withprobability one in the weighted measure, R ∗ r − = 0 .Proof. The relation (63) follows immediately from Itˆo’s formula. For the secondclaim, we note that in the unweighted measure we have R ∗ r − = 0. Since V isdecreasing in | x | , the additional drift given by the weighting points toward theorigin. (cid:3) Proposition 5.8.
Suppose γ is a simple curve in S r from to z with T > τ r .Then, M τ − = Λ c / τ − ∈ (0 , ∞ ) . LE IN MULTIPLY CONNECTED DOMAINS 47
Proof.
Easy estimates show that under the assumptions, Q τ − = 1, r ( τ − ) = 0, R τ − = 0. Proposition 5.2, then gives V τ − = 1. The assumptions also imply thatdist( γ τ , ˜ γ τ ) >
0, which implies 0 < Λ τ − < ∞ . (cid:3) Proposition 5.9. O t , t < τ r ∧ T is a local martingale with respect to P ∗ . Inparticular, M t = N t O t , t < τ r ∧ T is a local martingale with respect to P , and J t M t is a local martingale with respect to ˆ P .Proof. This is a restatement of the previous proposition in terms of the originalparametrization. (cid:3)
Let P ′ denote the probability measure obtained from weighting by the localmartingale M t . Proposition 5.10.
With P ′ probability one, τ r < T and (64) M = V ( r, x ) , M τ r = Λ c / τ r , In particular, V ( r, x ) = M = E [ M τ r ] = E h Λ c / τ r i . Proof.
The drift given by weighting by this martingale has a stronger drift to theorigin than for locally chordal
SLE κ and we know that that the latter one is good. (cid:3) ♣ There is a general principle that is being used here that is worth stressing. Suppose M t is a positive local martingale for t < τ . The martingale convergence theorem impliesthat with probability one the limit M τ = lim t → τ − M t exists. However, one cannot conclude E [ M ] = E [ M τ ] without more assumptions. One way to establish this equality is to considerthe paths weighted by the local martingale. If M τ exists and is finite with probability one inthe new measure , then we have uniform integrability and E [ M ] = E [ M τ ] . In our case weestablish that in the new measure we have R ∗ r − = 0 . If the latter fact holds, then we use aneasy deterministic estimate about curves to see that M τ < ∞ . ♣ At this point of the paper, the argument went very quickly, so it is a good idea to explainwhat has happened. The goal was to estimate the expectation (with respect to chordal
SLE κ in S r from to z ) of a random variable which is the exponential of the measure of a certainset of bad loops. For a curve γ and a loop l , we say that l is bad if l intersects γ , say atfirst time s ′ , but also intersects ˜ γ at first time s < s ′ . Suppose we have seen γ t . Then wecan split the bad loops into three sets: those with s < s ′ ≤ t ; those with s < t < s ′ ; andthose with t < s < s ′ . When we weight only by the first two sets of loops, we get the localmartingale N t , and the probability measure is locally chordal SLE κ . Lemma 3.15 shows thatthis is supported simple curves with γ ∩ ˜ γ = ∅ . We then weight again to include the third setof loops and this leads to the function V . Since we can show directly that V is decreasingin | x | (and here we were lucky with the monotonicity proved in Proposition 3.4), we can seethat the extra drift given by weighting by these loops points towards the origin and hence thismeasure is also supported simple curves with γ ∩ ˜ γ = ∅ . This allows us to justify the equation E [ M ] = E [ M τ r ] . Annulus
SLE κ from to x in S r The same ideas can be used to analyze ν S r (0 , x ) where 0 < | x | < π . For ease,we will assume x >
0, but the x < x ; if | x | ≥ π and γ connects 0 and x , then η = ψ ◦ γ is not simple. Asbefore, we define the measure by giving the Radon-Nikodym derivative as in (55) dν S r (0 , x ) dµ S r (0 , x ) ( γ ) = Y ( γ ) = 1 { γ ∩ ˜ γ = ∅} exp n c m ( γ ) o . The relevant functions are the following.(65) ˜ A ( r, x ) = π r X k ∈ Z \{ } sinh ( πx/ r )sinh ( π k/r ) sinh ( π ( x − πk ) / r ) , (66)˜ H I ( r, x ) = π r coth (cid:16) πx r (cid:17) + π r ∞ X k =1 (cid:20) coth (cid:18) π ( x + 2 π )2 r (cid:19) + coth (cid:18) π ( x − π )2 r (cid:19)(cid:21) . (67) ˜ L ( r, x ) = − ∂ x H S r (0 , x ) b H S r (0 , x ) = πr coth (cid:16) πx r (cid:17) . Lemma 6.1. If y ∈ R and f ( x ) = sinh x sinh ( x − y ) + sinh x sinh ( x + y ) , then f is increasing for ≤ x < y .Proof. Since f ( x ) = cosh(2 x ) − x − y ) − x ) − x + 2 y ) − , it suffices to show for every y ∈ R , that F ( x ) = cosh x − x − y ) − x − x + y ) − , is increasing for 0 ≤ x < y . Using the sum rule, we getcosh( x − y ) − x + y ) − x cosh y − , Letting r = cosh y ≥
1, we get[cosh( x − y ) −
1] [cosh( x + y ) −
1] = (cosh x cosh y − − sinh x sinh y = ( r cosh x − − ( r − x − x − r cosh x + r = (cosh x − r ) . Therefore, F ( x ) = 2 r (cosh x − r − ) (cosh x − x − r ) = 2 r e G (cosh x ) , where G ( t ) = log( t − r ) + log( t + 1) − t − r ) . LE IN MULTIPLY CONNECTED DOMAINS 49
Since r ≥ G ′ ( t ) > < t < r and hence G and F are increasing. (cid:3) Definition
The function ˜ V ( r, x ) , ≤ r < ∞ , < x < π is defined by(68) ˜ V ( r, x ) = E x (cid:20) exp (cid:26) − b Z σ ˜ A ( r − s, X s ) ds (cid:27)(cid:21) , where X t , ≤ t < σ satisifes(69) dX t = h H I ( r − t, X t ) − bκ ˜ L ( r − t, X t ) i dt + √ κ dB t , with B t is a standard Brownian motion and σ = inf { t : X t = 0 } . We define V ( r,
0) = 1.An important observation is that if X t satisfies (69) with X ∈ [0 , π ), then withprobability one σ < r and X t ∈ [0 , π ) for 0 ≤ t ≤ σ . Hence this is well defined.The function ˜ V , < r < ∞ , < x < π (70) ˙˜ V ( r, x ) = − b ˜ A ( r, x ) ˜ V ( r, x )+ h H I ( r, x ) − bκ ˜ L ( r, x ) i ˜ V ′ ( r, x )+ κ V ′′ ( r, x ) , where dot refers to r -derivatives and primes refer to x -derivatives.The definition of µ A r (1 , e iθ ) for 0 < θ < π takes a little more thought. Wewrite µ A r (1 , e iθ ) = µ A r (1 , e iθ ; R ) + µ A r (1 , e iθ ; L )where µ A r (1 , e iθ ; R ) denotes µ A r (1 , e iθ ) restricted to curves η such that the originlies in the component of D \ η whose boundary includes ( e iθ , dµ A r (1 , e ix ) d [ ψ ◦ ν S r (0 , x )] ( η ) = e br exp n − c m ∗ ( r, η ) o , where m ∗ ( r, η ) denotes the measure of the set of loops in A r of nonzero windingnumber that intersect η . Unlike the crossing case, the quantity on the right handside depends on η . It is not hard to give an expression for this. Let ˜ A denote thecomponent of A r \ η that contains C r on its boundary. let r γ = r γ,r be such that˜ A is conformally equivalent to A r γ . Then m ∗ ( r γ ) denotes the measure of loops ofnonzero winding number in ˜ A and hence m ∗ ( r, η ) = m ∗ ( r ) − m ∗ ( r γ ) . We could have also defined µ A r (1 , e iθ ) by dµ A r (1 , e iθ ) dµ D (1 , e iθ ) ( γ ) = 1 { γ ⊂ A r } exp n c m D ( γ, D \ A r ) o . Since these both satisfy (48), they must give the same measure.There is a subtlety that is worth mentioning. Let J denotes the closed diskabout 0 of radius e − r so that A r = D \ J and f : A r → D ⊂ D is a conformaltransformation that sends ∂ D to ∂ D . Informally we can write f ( J ) = K where D = D \ K , but the conformal map f is not defined on J . If z, w ∈ ∂ D , then f ◦ µ A r ( z, w ) = | f ′ ( z ) | b | f ′ ( w ) | b µ D ( f ( z ) , f ( w )) . This gives one way to construct µ D ( f ( z ) , f ( w )). But we also define it by theRadon-Nikodym derivative. Suppose γ ⊂ A r , then f ◦ γ ⊂ D and dµ A r ( z, w ) dµ D ( z, w ) ( γ ) = exp n c m D ( γ, J ) o ,dµ D ( f ( z ) , f ( w )) dµ D ( f ( w ) , f ( w )) ( f ◦ γ ) = exp n c m D ( f ◦ γ, K ) o . However, since f is not a conformal transformation of the disk, we have no reasonto believe that m D ( γ, J ) = m D ( f ◦ γ, K ) . Annulus
SLE κ in A r In the last section we considered the measure ν S r (0 , x + ir ) which we calledannulus SLE κ in the strip S r . This was analyzed by comparing the measure tochordal SLE κ in S r . Recall from (56) that the measure on paths given by annulus SLE κ restricted to a particular winding number is ν A r (1 , x ) = e br e − c m ∗ ( r ) / ψ ◦ ν S r (0 , x + ir ) . The term e br = | ψ ′ ( x + ir ) | b comes from conformal covariance and m ∗ ( r ) is theBrownian loop measure of loops in A r of nonzero winding number. Annulus SLE κ in A r from 1 to e − r + iθ is obtained from summing over all winding numbers µ A r (1 , e − r + iθ ) = X k ∈ Z ν A r (1 , θ + 2 πk ) . In this section we will compare ν A r (1 , x ) and µ A r (1 , e − r + iθ ) to to radial SLE κ in order to derive PDEs for the annulus partition functions. We will rederive anequation from [25].7.1. The differential equation.
Let ˜Ψ( r, x ) = | ν S r (0 , x + ir ) | be as in the pre-vious section, and let ˆ F ( r, x ) and F ( r, x ) denote the partition functions associatedto annulus SLE κ and annulus SLE κ restricted to a particular winding number,respectively. In other words, F ( r, x ) = | ν A r (0 , x ) | = β ( r ) ˜Ψ( r, x ) , where β ( r ) = exp (cid:26) br − c m ∗ ( r )2 (cid:27) = e br e − c r/ exp (cid:26) c Z r δ ( s ) ds (cid:27) , and ˆ F ( r, x ) = Ψ A r (1 , e − r + ix ) = ∞ X k = −∞ F ( r, x + 2 πk ) . Since(71) F ( r, x ) = β ( r ) ˜Ψ( r, x ) ≤ β ( r ) Ψ S r (0 , e x + ir ) ≍ β ( r ) r − b h cosh (cid:16) πx r (cid:17)i − b , we see that ˆ F ( r, x ) < ∞ . Recall the functions J and H I from Section 3.2. Asbefore, we will use dot for r -derivatives and primes for x -derivatives. LE IN MULTIPLY CONNECTED DOMAINS 51
Proposition 7.1. F satisfies the differential equation (72) ˙ F = κ F ′′ + H I F ′ + (cid:20) b H ′ I + b + ˜ b (6Γ( r ) − − br (cid:21) F. Moreover, ˆ F satisfies the same equation. ♣ As in [25], we check that this is consistent with what we know about κ = 2 for which b = 1 , ˜ b = 0 . For κ = 2 , from arguments based on the loop-erased walk we know that the SLE partition function for any domain D should be given by a multiple of the excursionPoisson kernel, H ∂D ( z, w ) . Hence a solution to (72) should be ˆ F ( r, x ) = H ∂A r (1 , e − r + ix )= e r X k ∈ Z H ∂S r (0 , x + 2 πk + i )= 12 e r J ( r, x ) . If this is so, then Proposition 7.1 implies that if Φ( r, x ) = 2 re − r ˆ F ( r, x ) = r J ( r, x ) , then ˙Φ = Φ ′′ + H I Φ ′ + H ′ I Φ . But we noted this relation in (11) . We set α ( r ) = b + ˜ b [6Γ( r ) −
1] = b − ˜ b + (2 b + c ) Γ( r ) , Θ( r, x ) = Θ κ ( r, x ) = H ′ I ( r, x ) + α ( r ) b − r , which allows us to write (72) as(73) ˙ F = κ F ′′ + H I F ′ + b Θ F. We will establish (73) for F . We note that F ( r, x ) is C in r and C in x . Indeed,in the previous section we showed the same for ˜Ψ( r, x ), and it is easy to show that m ∗ ( r ) is continuous in r and hence β ( r ) is differentiable. Hene we can use Itˆo’sformula freely. Before proceeding, let us show that this will also imply the resultfor ˆ F . Let X ( r ) t , ≤ t ≤ r , denote a solution to the SDE(74) dX ( r ) t = H I ( r − t, X ( r ) t ) dt + √ κ dB t . Then (73) and the Feynman-Kac formula implies that for r > t > F ( r, x ) = E x (cid:20) F ( r − t, X ( r ) t ) exp (cid:26) b Z t Θ( r − s, X ( r ) s ) ds (cid:27)(cid:21) , where E x denotes expectations assuming X ( r )0 = x . (We do not need to considerthe delicate case t = r so the conditions for the Feynman-Kac formula are easilyverified.) Using this and (71), we can see thatˆ F ( r, x ) = E x (cid:20) ˆ F ( r − t, X ( r ) t ) exp (cid:26) b Z t Θ( r − s, X ( r ) s ) ds (cid:27)(cid:21) , and by invoking the Feynman-Kac theorem again, we see that ˆ F also satisfies (73).To prove the proposition for F we compare radial SLE κ (from 1 to 0 in D )and annulus SLE κ (from 1 to e − r + iθ in A r ) for κ = 2 /a ≤
4. These measures,restricted to an initial segment of the path which has not reached C r , are absolutelycontinuous. It is useful to view radial
SLE κ raised onto the covering space H as we nowdescribe. We describe radial SLE κ as a periodic function on H . We return to theradial Loewner equation (17) which we write as(75) ∂G t ( z ) = a ( G t ( z ) − U t ) , G ( z ) = z, and view as an equation on H . Here U t is a standard Brownian motion with U = 0and cot ( z ) = cot( z/ γ in H such that withprobability one, for all t , γ t ∩ ˜ γ t = ∅ . Let η t = ψ ◦ γ t and define ˜ g t by˜ g t ( e iz ) = e iG t ( z ) . Then ˜ g t is the unique conformal transformation of D \ η t onto D with ˜ g t (0) =0 , ˜ g ′ t (0) >
0. In fact, ˜ g ′ t (0) = e at/ . Radial SLE is usually described in terms of thedifferential equation for ˜ g t .We now relate the equation (75) to the annulus Loewner equation described inSection 3.9. We fix an “initial radius” r . As in that section, we define r ( t ) and h t by saying that h t : S r \ ˆ γ t → S r ( t ) is a conformal transformation satisfying h t ( z +2 π ) = h t ( z )+2 π with h t ( ±∞ ) = ±∞ and h t ( γ ( t )) = U t . Recall that ∂ t h t ( z ) = 2 ˙ r ( t ) H r ( t ) ( h t ( z ) − U t ) . We define Φ t by h t = Φ t ◦ G t , and define ˜ h t , ˜Φ t by ˜ h t ( e iz ) = e ih t ( z ) , ˜ φ t ( e iz ) = e i Φ t ( z ) , so that ˜ h t = ˜ φ t ◦ ˜ g t . Note that ˜ h t is the unique conformal transformation of A r \ η t onto A r ( t ) with ˜ h t ( η ( t )) = e iU t . Also, for real x , | ˜ φ ′ t ( e ix ) | = Φ ′ t ( x ) . We note that (12) implies that for r ( t ) ≥ x ∈ R , | Φ ′ t ( x ) | = 1 + O ( e − r ( t ) ) , | Φ ′′ t ( x ) | = O ( e − r ( t ) ) . As in that section, we let σ s = inf { t : r ( t ) = s } , h ∗ s = h σ s , and we set ˜ h ∗ s = ˜ h σ s , ˜ φ ∗ s = ˜ φ σ s , ˜ g ∗ s = ˜ g σ s . Lemma 7.2.
Under the assumptions above, ∂ t | ˜ φ ′ t (1) | | t =0 = ∂ t Φ ′ t (0) | t =0 = a (cid:20) Γ( r ) − r (cid:21) ,∂ s | ( φ ∗ r − s ) ′ (1) | | s =0 = 2 Γ( r ) − r . Here Γ( r ) is as defined in (20) . LE IN MULTIPLY CONNECTED DOMAINS 53
Proof.
Note that a ( z ) = a (cid:20) z − z (cid:21) + O ( | z | ) . Recall that ˙ r (0) = − a/ − a H r ( z ) = a (cid:20) z + z (cid:18) Γ( r ) − − r (cid:19)(cid:21) + O ( | z | ) , Therefore, the first result follows from (43) and the second from ˜ φ t = ˜ φ ∗ r ( t ) . (cid:3) Let µ , µ , µ denote µ D (1 , − , µ D (1 , , and ν A r (1 , x ), respectively, and let w = e − r + ix . Let z t = e iU t = ˜ g t ( η ( t )) , ζ t = ˜ g t ( − , w t = ˜ g t ( w ) , x t = arg w t , where x t is chosen to be continuous in t with x = x . If t < τ r , these three measuresare absolutely continuous with respect to each other and we can write down theRadon-Nikodym derivatives. Recall from Section 3.8 that dµ dµ ( η t ) = ˜ g ′ t (0) ˜ b Ψ D ( z t , | ˜ g ′ t ( − | b Ψ D ( z t , ζ t ) = ˜ g ′ t (0) ˜ b | ˜ g ′ t ( − | b Ψ D ( z t , ζ t ) . Using similar reasoning for annulus
SLE with respect to chordal
SLE , we get dµ dµ ( η t ) = | ˜ g ′ t ( w ) | b | ν ˜ g t ( A r ) ( z t , x t ) | exp (cid:8) c m D ( D r , η t ) (cid:9) | ˜ g ′ t ( − | b Ψ D ( z t , ζ t ) . We have not actually defined the measure ν ˜ g t ( A r ) ( z t , x t ), so let us describe it now.Since ˜ g t ( A t ) is a conformal annulus whose outer boundary is the unit circle, we candefine ν ˜ g t ( A r ) ( z t , x t ) in the same way that ν A r (1 , x ) was defined. In other words,it is annulus SLE between z t and w t in the conformal annulus ˜ g t ( A r ) restricted tocurves of a particular winding number. The choice of winding number is determinedby continuity in t .Let M t = dµ dµ ( η t ) = ˜ g ′ t (0) − ˜ b | ˜ g ′ t ( w ) | b | ν ˜ g t ( A r ) ( z t , x t ) | exp n c m D ( D r , η t ) o . We see that M t is a local martingale for radial SLE κ . Let ˜ h t = ˜ φ t ◦ ˜ g t . Conformalcovariance implies that | ν ˜ g t ( A r ) ( z t , x t ) | = | ˜ φ ′ t ( e iU t ) | b | ˜ φ ′ t (˜ g t ( w )) | b Ψ A r ( t ) ( e iU t , ˜ φ t (˜ g t ( w ))) . Therefore, M t = ˜ g ′ t (0) − ˜ b | ˜ φ ′ t ( e iU t ) | b exp n c m D ( D r , η t ) o | ˜ h ′ t ( w ) | b F ( r ( t ) , R t ) , where R t = Re[ h t ( z ) − U t ] . We have shown the following.
Proposition 7.3. If U t is a standard Brownian motion, then M t = J ( t ) F ( r ( t ) , R t ) , is a local martingale where J ( t ) = ˜ g ′ t (0) − ˜ b | ˜ φ ′ t ( e iU t ) | b exp n c m D ( D r , η t ) o | ˜ h ′ t ( w ) | b , and R t = Re[ h t ( x + ir ) − U t ] . Using this proposition, we can write down a differential equation for F ( s, x ).It is convenient to write the local martingale in the annulus parametrization. Let U ∗ s = U σ r − s . Then U ∗ s is a martingale with quadratic variation σ r − s . Let R ∗ s = h ∗ r − s ( z ) − U ∗ s . Then dR ∗ s = ∂ s h ∗ r − s ( z ) dt + dU ∗ s Note that ∂ s h ∗ r − s ( z ) = H I ( r, x ) , ∂ s σ r − s | s = r = 2 /a = κ. The last proposition becomes the following.
Proposition 7.4.
For fixed r > , if R ∗ s = h ∗ r − s ( z ) − U ∗ s and M ∗ s = J ∗ ( r − s ) F ( r − s, R ∗ s ) , where J ∗ ( s ) = (˜ g ∗ s ) ′ (0) − ˜ b | ˜( φ ∗ s ) ′ ( e iU σs ) | b exp n c m D ( D s , η σ s ) o | (˜ h ∗ s ) ′ ( w ) | b , then M ∗ s is a martingale. If we write dots for r -derivatives, then by considering the martingale at time s = 0 and using Itˆo’s formula, we get the equation˙ F = κ F ′′ + H I F ′ − ˙ J F, where − ˙ J ( r ) = ∂ s J ( r − s ) | s =0 . All the remains for proving Proposition 7.1 is to calculate − ˙ J ( r ). Lemma 7.5. − ˙ J ( r ) = α ( r ) + b H ′ I ( r, x ) − br . Proof.
We have parametrized radial
SLE κ such that ∂ t ˜ g ′ t (0) = ( a/ g ′ t (0) , and hence ∂ t log h ˜ g ′ t (0) − ˜ b i | t =0 = − a ˜ b b (1 − a )4 ,∂ s log h (˜ g ∗ r − s ) ′ (0) − ˜ b i s =0 = − ˜ b. The relationship between the Brownian loop measure and the bubble measure im-plies ∂ s c m D ( D r , η σ r − s ) | s =0 = 2 a ∂ t c m D ( D r , η t ) | t =0 = c Γ D (1 , A r ) = c Γ( r ) . Lemma 7.2 shows that ∂ s log | ( ˜ φ ∗ r − s ) ′ ( U ∗ s ) | b | s =0 = − br + 2 b Γ( r ) . Recall that if z = x + ir, w = e iz = e − r + ix , ˜ h ∗ s ( w ) = e ih ∗ s ( z ) , and hence | (˜ h ∗ r − s ) ′ ( w ) | = e r e − Im[ h ∗ r − s ( z )] | ( h ∗ r − s ) ′ ( z ) | = e s | ( h ∗ r − s ) ′ ( z ) | . Therefore, using (42), we have ∂ s log | (˜ h ∗ r − s ) ′ ( w ) | b | s =0 = b + b H ′ I ( r, x ) . LE IN MULTIPLY CONNECTED DOMAINS 55
Adding all the terms, gives b − ˜ b + ( c + 2 b ) Γ( r ) + b H ′ I ( r, x ) − br = α ( r ) + b H ′ I ( r, x ) − br . (cid:3) Comparing annulus
SLE with radial
SLE large r . We now have anessentially complete description of annulus
SLE κ . In our framework, this is ameasure µ A r (1 , e − r + ix ) of total mass ˆ F ( r, x ). In the next subsection, we will provethe following. Theorem 7.6.
There exist c ∗ , q ∈ (0 , ∞ ) such that uniformly in x , ˆ F ( r, x ) = c ∗ r c / e ( b − ˜ b ) r [1 + O ( e − qr )] , r → ∞ . Let µ = µ D (1 ,
0) as before and let µ = µ A r (1 , e − r + ix ) with correspondingprobability measure µ . Suppose t is sufficiently small so that a curve starting atthe unit disk cannot reach C r by time t . Then, similarly to the previous section, if w = e − r + ix and ζ t = ˜ g t ( γ ( t )), we can write dµ dµ ( η t ) = | ˜ g ′ t ( w ) | b ˜ g ′ t (0) ˜ b exp n c m D ( D r , η t ) o | µ ˜ g t ( A r \ η t ) ( ζ t , ˜ g t ( w )) | . Proposition 7.7.
There exists q > such that uniformly over t > , r ≥ ta + 2 ,and all initial segments γ t , dµ dµ ( η t ) = c ∗ e r ( b − ˜ b ) r c / [1 + O ( e − qu )] , where u = r − ta . In particular, there exists c < ∞ such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ dµ ( η t ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − qu . Proof.
Let φ t : ˜ g t ( A r \ η t ) → A s be a conformal transformation sending C to C and let h t = φ t ◦ ˜ g t . Using conformal covariance, we write dµ dµ ( η t ) = | h ′ t ( w ) | b | φ ′ t ( ζ t ) | b ˜ g ′ t (0) ˜ b exp n c m D ( D r , η t ) o | µ A s ( h t ( w ) , φ t ( ζ t )) | . Suppose t is given, r ≥ ta +2 and let u = r − ta . Recall that in our normalization˜ g ′ t (0) = e at/ . Using the deterministic estimates from Lemma ?? , we get | h ′ t ( w ) | b = e atb/ [1 + O ( e − u )] , | φ ′ t ( ζ t ) | b = 1 + O ( e − u ) , exp n c m D ( D r , η t ) o = ( r/u ) c / [1 + O ( e − u )] ,s = u + O ( e − u ) , | µ A s ( h t ( w ) , φ t ( ζ t )) | = c ∗ u c / e ( b − ˜ b ) u [1 + O ( e − u )] . Combining these estimates gives the first equality and since the dominant factordoes not depend on the initial segment, the second equality follows. (cid:3)
Proof of Theorem 7.6.
Let λ ( r ) = r b exp (cid:26) − Z r α ( s ) ds (cid:27) ,K ( r, x ) = λ ( r ) F ( r, x ) ,K ( r, x ) = λ ( r ) Ψ A r (1 , e − r + ix ) = λ ( r ) ˆ F ( r, x ) = X k ∈ Z F ( r, x + 2 πk ) . Proposition 3.5 gives α ( r ) = b − ˜ b + (2 b + c ) Γ( r ) = b − ˜ b + 2 b + c + O ( e − r )2 r , and hence λ ( r ) = λ ∞ r − c / e (˜ b − b ) r [1 + O ( r − e − r )] . Therefore, to prove Theorem 7.6, it suffices to show that there exists K ∞ ∈ (0 , ∞ )and c < ∞ such that | K ( r, x ) − K ∞ | ≤ c e − r . Since ˙ λ ( r ) = λ ( r ) (cid:20) br − α ( r ) (cid:21) , it follows from Proposition 7.1 that K , K satisfy˙ K = κ K ′′ + H I K ′ + b H ′ I K , (76) ˙ K = κ K ′′ + H I K ′ + b H ′ I K. The Feynman-Kac representation tells us that if r > t > K ( r, x ) = E x (cid:20) K ( r − t, X ( r ) t ) exp (cid:26)Z t J ( r − s, X ( r ) s ) ds (cid:27)(cid:21) , where X ( r ) t satisfies (74). Recall that(78) | H I ( r, x ) | , | J ( r, x ) | ≤ c e − r , r ≥ , which implies(79) (cid:12)(cid:12)(cid:12)(cid:12)Z r − t H ′ I ( z, X ( r ) s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − t , exp (cid:26) b Z r − H ′ I ( z, X ( r ) s ) ds (cid:27) ≍ , and for r ≥ K ( r, x ) ≍ E x [ K ( X r − )] ≤ c E x [exp {− bX r − } ] , where X s = X ( r ) s . ♣ Those experienced with PDEs can probably skip the rest of this section. Since | H I | + | H ′ I | = O ( e − r ) , for large r the equation (76) is well approximated by the standard heatequation ˙ K = κ K ′′ . One just needs to keep track of the error terms. I have taken aprobabilistic approach using coupling, but this is just personal preference. We will use standard coupling techniques to analyze the equation. Here is thebasic estimate. We write x ≡ y if ( y − x ) / π ∈ Z . LE IN MULTIPLY CONNECTED DOMAINS 57
Lemma 7.8.
There exist u > , c < ∞ such that the following holds. Suppose r ≥ and X t = X ( r ) t , Z t = Z ( r ) t are independent solutions to (74) with X = x, Z = y with x ≤ y < x + 2 π . Let T = inf { t : X t ≡ Z t } . Then, P { T ≥ t } ≤ c e − ut , and if t ≤ , P { T ≥ t } ≤ c t − ( y − x ) . If we define Y t = (cid:26) Z t t < TZ T + ( X t − X T ) t ≥ t Then Y t satisfies (74) with Y = y and Y t ≡ X t for t ≥ T . Proposition 7.9. • There exist < c < c < ∞ such that (80) c ≤ K ( r, x ) ≤ c , r ≥ , x ∈ R . • There exists K ∞ ∈ (0 , ∞ ) and u > and c < ∞ such that | K ( r, x ) − K ∞ | ≤ c e − ur . Proof.
For fixed r , x ≤ y ≤ x + 2 π , let X t , Y t , T be as in Lemma 7.8 and let m − ( r ) , m + ( r ) be the minimum and maximum, respectively, of K ( r, x ) for 0 ≤ x ≤ π . From (77) and (79), we see that c m − (1) ≤ K ( r, x ) ≤ c m + (1) . Using (79), K ( r, x ) = E x (cid:2) F ( r/ , K r/ ) (cid:3) [1 + O ( e − r/ )] . This gives (80). Combining this with the coupling, we see that K ( r, x ) = K ( r, y ) (cid:2) O ( e − ur ) (cid:3) . (cid:3) References [1] Robert O. Bauer and Roland M. Friedrich. Stochastic Loewner evolution in multiply con-nected domains.
C. R. Math. Acad. Sci. Paris , 339(8):579–584, 2004.[2] V. Beffara. The dimension of the SLE curves.
Annals of probability , 36(4):1421–1452, 2008.[3] Shawn Drenning. Excursion reflected Brownian motion. In preparation.[4] Julien Dub´edat. SLE and the free field: partition functions and couplings.
J. Amer. Math.Soc. , 22(4):995–1054, 2009.[5] Avner Friedman.
Stochastic differential equations and applications . Dover Publications Inc.,Mineola, NY, 2006. Two volumes bound as one, Reprint of the 1975 and 1976 original pub-lished in two volumes.[6] Ioannis Karatzas and Steven E. Shreve.
Brownian motion and stochastic calculus , volume113 of
Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1991.[7] Yˆusaku Komatu. On conformal slit mapping of multiply-connected domains.
Proc. JapanAcad. , 26(7):26–31, 1950.[8] Michael J. Kozdron and Gregory F. Lawler. The configurational measure on mutually avoidingSLE paths. In
Universality and renormalization , volume 50 of
Fields Inst. Commun. , pages199–224. Amer. Math. Soc., Providence, RI, 2007.[9] G. Lawler. Schramm-Loewner evolution (SLE). In
Statistical mechanics , volume 16 of
IAS/Park City Math. Ser. , pages 231–295. Amer. Math. Soc., Providence, RI, 2009.[10] G.F. Lawler.
Conformally invariant processes in the plane . Amer Mathematical Society, 2008.[11] G.F. Lawler and S. Sheffield. A natural parameterization for the Schramm-Loewner evolution. to appear in Annals Probab. [12] Gregory Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordalcase.
J. Amer. Math. Soc. , 16(4):917–955 (electronic), 2003.[13] Gregory F. Lawler. The Laplacian- b random walk and the Schramm-Loewner evolution. Illi-nois J. Math. , 50(1-4):701–746 (electronic), 2006.[14] Gregory F. Lawler. Partition functions, loop measure, and versions of SLE.
J. Stat. Phys. ,134(5-6):813–837, 2009.[15] Gregory F. Lawler and Vlada Limic.
Random walk: a modern introduction , volume 123 of
Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010.[16] Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal invariance of planarloop-erased random walks and uniform spanning trees.
Ann. Probab. , 32(1B):939–995, 2004.[17] Gregory F. Lawler and Jos´e A. Trujillo Ferreras. Random walk loop soup.
Trans. Amer.Math. Soc. , 359(2):767–787 (electronic), 2007.[18] Gregory F. Lawler and Wendelin Werner. The Brownian loop soup.
Probab. Theory RelatedFields , 128(4):565–588, 2004.[19] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees.
IsraelJournal of Mathematics , 118(1):221–288, 2000.[20] O. Schramm and S. Rohde. Basic properties of SLE.
Annals of mathematics , 161(2):883,2005.[21] Oded Schramm and Scott Sheffield. Contour lines of the two-dimensional discrete Gaussianfree field.
Acta Math. , 202(1):21–137, 2009.[22] S. Sheffield and W. Werner. Conformal loop ensembles: construction via loop-soups.
Arxivpreprint arXiv:1006.2372 , 2010.[23] Stanislav Smirnov. Conformal invariance in random cluster models. I. Holomorphic fermionsin the Ising model.
Ann. of Math. (2) , 172(2):1435–1467, 2010.[24] Dapeng Zhan. Reversibility of chordal SLE.
Ann. Probab. , 36(4):1472–1494, 2008.[25] Dapeng Zhan. Reversibility of whole-plane SLE.