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Let τ = τz be the supremum of the set of t such that gt(z) is well-defined. Then either τ = , or lim ր g (z) W = 0. Set H := z H : τ > t . It is z ∞ t τ t − t t { ∈ z } easy to check that for every t> 0 the map gt : Ht H is conformal. It has been →−1 shown [RS05, LSW04] that the limit γ(t) = limz→Wt gt (z) exists and is continuous in t. This is the path defined by SLE. It is easy to check that for every t 0, Ht is the unbounded connected component of H γ[0,t]. ≥ Radial SLE is defined in a similar manner,\ except that the upper half plane H is replaced by the unit disk D, and the differential equation (1) is replaced by g (z)+ W (2) ∂ g (z)= g (z) t t , g (z)= z, t t − t g (z) W 0 t − t where Wt is Brownian motion on the unit circle ∂D, with time scaled by a factor of κ. The definition is presented in detail and generalized in Section 2. Yet an- other version of SLE, dipolar SLE, was introduced in [BB04] and further studied in [BBH05]. In [LSW01a] it was shown that radial SLE and chordal SLE have equivalent (i.e., mutually absolutely continuous) laws, up to a time change, when stopped before the disconnection time (the precise meaning of this should become clear soon). Here, we carry this calculation a step further, and describe what a radial SLE looks like when transformed to the chordal coordinate system and vice versa. We extend the definition of the SLE(κ; ρ) processes from [LSW03, Dub05] to the setting where the force points are permitted to be in the interior of the domain. With the extended definitions, the SLE(κ; ρ) radial processes are transformed into appropriate SLE(κ; ρ) chordal processes, and vice versa. We start with the definition of chordal SLE(κ; ρ) with a force point in the interior. Definition 1. Let z H, w R, κ 0, ρ R. Let B be standard one- 0 ∈ 0 ∈ ≥ ∈ t dimensional Brownian motion. Define (Wt, Vt) to be the solution of the system of SDEs 1 dW = √κ dB + ρ dt, t t ℜW V t − t 2 dt dV = , t V W t − t starting at (W0, V0) = (w0,z0), up to the first time τ > 0 such that inf W V : t [0, τ) =0. {| t − t| ∈ } (Here, z denotes the real part of z.) Then the solution of Loewner’s chordal ℜ equation (1) with Wt as the driving term will be called chordal SLE(κ; ρ) starting at (w0,z0). One motivation for this definition comes from the fact that, as we will later see, ordinary radial SLE(κ) starting at W0 = w is transformed by a M¨obius transfor- mation ψ : D H to a time changed chordal SLE(κ; κ 6) starting at (ψ(w), ψ(0)) (both up to a→ corresponding positive stopping time). − In [Wer04a] it was shown that a chordal SLE(κ; ρ) process can be viewed as an ordinary SLE(κ) process weighted by a martingale. In Section 5 we extend this to SLEs with force points in the interior of the domain. SLE Coordinate Changes 3
2. A unified framework for SLE(κ; ρ) In order to produce a clean and general formulation of the change of coordinate results, we need to introduce a bit of notation which will enable us to deal with the chordal and radial versions simultaneously. Let X D, H . We let ΨX (w,z) denote the corresponding Loewner vector field; that is,∈{ } z + w 2 ΨD(w,z)= z , ΨH(w,z)= . − z w z w − − Thus, Loewner’s radial [respectively chordal] equation may be written as
∂tgt(z)=ΨX Wt,gt(z) if X = D [respectively X = H]. Let ID denote the inversion in the unit circle ∂D, and let IH denote the inversion in the real line ∂H, i.e., complex conjugation. Set
Ψ (z, w)+Ψ (I (z), w) Ψ˜ (z, w) := X X X . X 2
In radial ordinary SLE, Wt = W0 exp(i√κBt), where Bt is a Brownian motion. By Itˆo’s formula, dW = (κ/2) W dt + i √κ W dB . On the other hand, in ordinary t − t t t chordal SLE, one simply has dWt = √κ dBt. We thus set
D(W , dB , dt) := (κ/2) W dt + i √κ W dB , H(W , dB , dt) := √κ dB . G t t − t t t G t t t
Definition 2. Let κ 0, m N, ρ1,ρ2,...,ρm R. Let X H, D and 1 2 m ≥ ∈ ∈ ∈ { } let V , V ,...,V X. (When X = H, we include in X.) Let w0 1 m∈ ∞ ∈ ∂X , V ,...,V and let Bt be standard one-dimensional Brownian motion. Consider\ {∞ the solution} of the SDE system
m ρ (3) dW = (W , dB , dt)+ j Ψ˜ (V j , W ) dt, t GX t t 2 X t t Xj=1 j j (4) dVt =ΨX (Wt, Vt ) dt, j =1, 2,...,m
j j starting at W0 = w0 and V0 = V , j = 1,...,m, up to the first time τ such that for some j inf W V j : t < τ = 0. Let g (z) be the solution of the ODE {| t − t | } t
∂tgt(z)=ΨX (Wt,gt(z)) starting at g (z)= z. Let K = z X : τ t , t < τ, be the corresponding hull, 0 t { ∈ z ≤ } defined in the same way as for ordinary SLE. Then the evolution t Kt will be 1 m 7→ called X-SLE(κ; ρ1,...,ρm) starting at (w0, V ,...,V ). When X = D, we refer D H j to -SLE as radial SLE, while -SLE is chordal. The points Vt will be called force points.
j j Note that Vt = gt(V ), for all t [0, τ). In some situations, it is possible∈ and worthwhile to extend the definition of SLE(κ; ρ1,...,ρm) beyond the time τ (see, e.g., [LSW03]), but we do not deal with this here. 4 Oded Schramm and David B. Wilson
3. Some M¨obius coordinate changes
Set oH := and oD := 0 and as before X H, D . Observe that if V m = o , ∞ ∈{ } X then the value of ρm has no effect on the X-SLE(κ; ρ1,...,ρm), and, in fact, the SLE(κ; ρ1,...,ρm) reduces to SLE(κ; ρ1,...,ρm−1). Consequently, by increasing m and appending oX to the force points, if necessary, there is no loss of generality in assuming that m ρ = κ 6. This assumption simplifies somewhat the statement j=1 j − of the followingP theorem: Theorem 3. Let X, Y D, H , and let ψ : X Y be a conformal home- ∈ { } → omorphism; that is, a M¨obius transformation satisfying ψ(X) = Y . Let w0 1 2 m ∈ ∂X , V , V ,...,V X w0 . Suppose that ρ1,ρ2,...,ρm R sat- \ {∞}m ∈ \{ } ∈ isfy j=1 ρj = κ 6. Then the image under ψ of the X-SLE(κ; ρ1,...,ρm) −1 m startingP from (w0, V ,...,V ) and stopped at some a.s. positive stopping time has the same law as a time change of the Y -SLE(κ; ρ1,...,ρm) starting from 1 m (ψ(w0), ψ(V ),...,ψ(V )) stopped at an a.s. positive stopping time. The stopping time for the X-SLE may be taken as the minimum of τ and the −1 first time t such that ψ (oY ) Kt. The case when X = Y = H and∈ the force points are on the boundary appears in [Dub04].
Proof. Suppose first that X = H and Y = D. Let gt be the family of maps gt : H Kt H. Let φt : H D be the M¨obius transformation such that \ → −1 → ′ Ft := φt gt ψ satisfies the radial normalization Ft(0) = 0 and Ft (0) (0, ). ◦ ◦ −1 ∈ ∞ Let zt = xt + iyt := gt ψ (0), with xt,yt R. There is some (unique) λt ∂D such that ◦ ∈ ∈ z zt φt(z)= λt − . z zt ′ − Let s(t) := log Ft (0) and let t(s) denote the inverse of the map t s(t). By the chain rule, 7→
′ ′ λt ′ −1 −1 ′ s (t)= ∂t log Ft (0) = ∂t log gt(ψ (0)) (ψ ) (0) 2iyt = ∂ log g′(ψ−1(0)) ∂ log y . ℜ t t − t t We have by (1) 2 y ∂ y = − t , t t z W 2 | t − t| and 2 2 g′(z) ∂ g′(z)= ∂ ∂ g (z)= ∂ = − t . t t z t t z g (z) W (g (z) W )2 t − t t − t Using this with z = o in our previous expression for s′(t) gives 2 2 4 y2 s′(t)= − + = t . ℜ(z W )2 z W 2 z W 4 t − t | t − t| | t − t| ˆ ˆ Setg ˆs(z) := Ft(s)(z), Ws := φt(s)(Wt(s)) and Ks := ψ(Kt(s)). We need to verify that Loewner’s equation holds:
(5) ∂ gˆ (z)=ΨD Wˆ , gˆ (z) , z D Kˆ . s s s s ∈ \ s SLE Coordinate Changes 5
This may be verified by brute-force calculation or may be deduced from Loewner’s ′ s theorem, as follows: first,g ˆs is appropriately normalized:g ˆs(0) =0 andg ˆs(0) = e . Second, the domain of definition ofg ˆs is clearly D Kˆs. Lastly, if ε > 0 is small, −1 H \ then gt+ε gt is defined on the complement in of a small neighborhood of Wt. Consequently,◦ g ˆ gˆ−1 is defined on the complement in D of a small neighborhood s+ε ◦ s of Wˆ s when ε> 0 is small. Therefore, Loewner’s theorem gives (5). It remains to calculate dWˆ . Since Wˆ = λ Wt−zt , the Itˆodifferential d log Wˆ s s t Wt−zt s satisfies (6) d log Wˆ = d log λ + d log(W z ) d log(W z ). s t t − t − t − t Sinceg ˆs(ψ( )) = φt(s)( ) = λt(s), Equation (5) with z = ψ( ) gives ∂sλt(s) = ˆ ∞ ∞ ∞ ΨD(Ws, λt(s)), so
−1 −1 A := d log λ = λ ΨD(Wˆ , λ ) ds = Wˆ ΨD(λ , Wˆ ) ds. 0 t t s t − s t s Itˆo’s formula gives (7) κ dt dW dz d log(W z )= + t − t t − t −2(W z )2 W z t − t t − t κ dt dW 2 dt (4 κ) dt dW = + t + = − + t . −2(W z )2 W z (z W )2 2(W z )2 W z t − t t − t t − t t − t t − t Likewise, (4 κ) dt dW (8) d log(W z )= − t . − t − t −2(W z )2 − W z t − t t − t We now handle the sum of the dt terms in (7) and (8), using our above expression for s′(t). (4 κ) dt (4 κ) dt 2 i (4 κ) y (W x ) s′(t)−1 ds A := − − = − t t − t 1 2(W z )2 − 2(W z )2 W z 4 t − t t − t | t − t| i (4 κ) (W x ) ds = − t − t 2 yt 4 κ −1 = − Wˆ ΨD(λ , Wˆ ) ds. − 2 s t s
Next, we look at the dWt terms. First, write (3) in slightly different form
2m ρj j dW = √κ dB + ΨH(v , W ) dt, t t 4 t t Xj=1 j j j+m j where vt := Vt , vt := IH(vt ) and ρj+m = ρj for j =1, 2,...,m. The sum of the dWt terms in (7) and (8) is 2m dWt dWt 2 iyt ρj j A := = √κ dB + ΨH(v , W ) dt 2 W z − W z W z 2 t 4 t t t − t t − t | t − t| Xj=1 2m ′ −1/2 ρj j = i κs′(t) dB + is (t) ΨH(v , W ) ds. t 4 t t p Xj=1 6 Oded Schramm and David B. Wilson
Now define t(s) ′ Bˆs := s (t) dBt. Z0 p ˆ t(s0) ′ ˆ Then B s0 = 0 s (t) dt = s0, and therefore s Bs is standard Brownian h i j j 7→ motion. Also setR v ˆ := φ (v ), and note that φ (IH(z)) = ID(φ (z)), since φ : H t t t t t t → D is a M¨obius transformation. To further translate our expression for A2 to the D coordinate system, we calculate
2 ′ −1/2 j i zt Wt ˆ −1 j ˆ ˆ −1 ˆ is (t) ΨH(vt , Wt)= | − | = Ws ΨD(ˆvt , Ws) Ws ΨD(λt, Ws). y (W vj ) − t t − t Consequently, since m ρ = κ 6, j=1 j − P m κ 6 −1 −1 ρj j A = i √κ dBˆ − Wˆ ΨD(λ , Wˆ ) ds + Wˆ Ψ˜ D(ˆv , Wˆ ) ds. 2 s − 2 s t s s 2 t s Xj=1
Setttingw ˆs := log Wˆ s, we get
m −1 ρj j dwˆ = A + A + A = i √κ dBˆ + Wˆ Ψ˜ D(ˆv , Wˆ ) ds. s 0 1 2 s s 2 t s Xj=1 Now, Itˆo’s formula gives
1 dWˆ = d exp(w ˆ ) = exp(ˆw ) dwˆ + exp(w ˆ ) d wˆ s s s s 2 s h si m ˆ ρj j κ Ws = i √κ Wˆ dBˆ + Ψ˜ D(ˆv , Wˆ ) ds ds. s s 2 t s − 2 Xj=1 This completes the proof in the case X = H and Y = D. If X = D and Y = H, we may just reverse the above equivalence. To handle the case X = Y = H, we may just write the M¨obius trasformation ψ : X Y as a → composition ψ = ψ2 ψ1, where ψ1 : H D and ψ2 : D H, and appeal to the above situtations. The◦ case X = Y = D →is similar. →
Remark 4. It is possible to come up with a somewhat more conceptual version of some parts of this proof. First, the time change s′(t) is the rate at which the radial capacity changes with respect to the chordal capacity. This is known to be ′ 2 ˆ φt(Wt) . Second, we have Ws = φt(Wt). Itˆo’s formula gives ˆ ′ ′′ dWs = (∂tφt)(Wt) dt + φt(Wt) dWt + (κ/2) φt (Wt) dt.
′ j The terms φt(Wt)ΨH(Vt , Wt) that arise by expanding dWt should be thought of as j ˆ the image under φt of the vector field z ΨH(Vt ,z), evaluated at Ws. The vector j 7→ field z ΨH(V ,z) is the Loewner vector field in H, and it should be mapped 7→ t under φt to a multiple of the Loewner vector field in D plus some vector field which preserves D. SLE Coordinate Changes 7
4. Strip SLE For comparison and illustration, we now mention another type of SLE. It gen- eralizes dipolar SLE as introduced in [BB04] and studied in [BBH05] as well as a version of dipolar SLE with force points, which was used in [LSW01b, 3] § Definition 5 (Dipolar SLE). Let S = x + iy : x R, y (0,π/2) . Set { ∈ ∈ } ΨS(w,z) = 2 coth(z w), IS = IH, Ψ (z,w)+Ψ (−I (z),w) Ψ˜ (z, w)= S S S , = H. S 2 GS G Then strip-SLE(κ; ρ1,ρ2,...,ρm) is defined using Definition 2 with X = S and w0 R. In the case where m = 0, this coincides with dipolar SLE, as defined in [BBH05].∈ Note that in strip-SLE(κ; ρ ,...,ρ ), possible force points at + and at 1 m ∞ −∞ do exert an effect on the motion of Wt, since ΨS( , w) = 2. However, if m−1 m ±∞ ∓ V = + and V = , then adding a constant to both ρm−1 and ρm has no effect,∞ since the resulting−∞ forces cancel. Therefore, we may again assume with no loss of generality that m ρ = κ 6. In that case, the image of this j=1 j − process under any conformal mapP ψ : S H that satisfies ψ(w0) = is chordal- → 6 ∞ SLE(κ; ρ1,ρ2,...,ρm). The proof is left to the dedicated reader.
5. Associated martingales In [Wer04a] it was shown that a chordal SLE(κ; ρ) process can be viewed as an ordinary SLE(κ) process weighted by a martingale. In the following, we extend this to most of the SLE-like processes discussed in the previous sections. Suppose that (Y ,t 0) is some random process taking values in some space t ≥ X. If h : X [0, ) is some function such that h(Y1) is measurable and 0 < E[h(Y )] < ,→ then∞ we may weight our given probability measure by h(Y ); that 1 ∞ 1 is, we may consider the probability measure P˜ whose Radon–Nykodim derivative with respect to P is h(Y1)/E[h(Y1)]. In some cases, the new law of Yt is called the Doob-transform of the unweighted law. In many situations one can explicitly determine the Doob-transform. Consider, for example, a diffusion process Yt adapted to the filtration t taking values in some domain in Rn. If h is as above, then M := E h(Y ) Fis a martingale. It t 1 Ft turns out to be worthwhile to forget about h and consider weighting by any positive martingale. Indeed, for every event A we have P˜ [A] = E 1 M /M . Gir- ∈ Ft A t 0 sanov’s theorem (see, e.g. [RY99]) describes the behavior of contin uous martingales Nt weighted by a positive continuous martingale Mt adapted to the same filtration t. It states that F t d N,M t′ Nt h i − Z Mt′ is a local martingale under the weighted measure. In particular, if Bt is Brownian motion under P and dMt = at dBt, for some adapted process at, then B˜t := t ′ B (a ′ /M ′ ) dt is a P˜ -Brownian motion (note that B˜ = B ). Thus, with t − t t h it h it respectR to P˜, Bt has the drift term (at/Mt) dt. 8 Oded Schramm and David B. Wilson
Theorem 6. Consider standard chordal SLE(κ). Let z1,...,zn be some collection H R j j j of distinct points in . Let ρ1,...,ρn be arbitrary. Define zt = xt + iyt := j ∈ gt(z ). Set n ′ − j 2 M := g (zj) (8 2κ+ρj )ρj /(8κ)(y )ρj /(8κ) W z ρj /κ t | t | t | t − t| × jY=1 ′ ′ ρ ρ ′ /(4κ) zj zj zj zj j j . t − t t − t 1≤j Then Mt is a local martingale. Moreover, under the measure weighted by Mt (with an appropriate stopping time) we have chordal SLE(κ; ρ1,...,ρn) with force points z1,...,zn. The local martingale Mt was independently discovered by Marek Biskup [Bis04]. Proof. Using Itˆo’s formula, we may calculate n n 1 ρ 1 ρ dM = M j dW = M j dB . t t κ ℜ W z t t √κ ℜ W z t Xj=1 t − t Xj=1 t − t This is laborious, but straightforward. Thus, Mt is a local martingale. The drift term d W, M h it Mt from Girsanov’s theorem is thus precisely the drift one has for SLE(κ; ρ1,...,ρn) (since it is the product of the coefficient of dBt in dMt and in dWt divided by Mt). Remark 7. A similar variation of the theorem holds when some or all of the points j j ′ j z are on the real axis. One only needs to replace the corresponding yt by gt(z ) in the definition of Mt. We now discuss the radial version of the theorem. Let z1,...,zn be distinct points in the unit disk. We may extend the radial maps gt to the complement n+j j of the unit disk by Schwarz reflection, gt(ID(z)) = ID(gt(z)). Set z := ID(z ), j j ρn+j := ρj and zt = gt(z ) for j =1,..., 2 n. Then the local martingale takes the form 2n (9) M = g′(0)q0 W zj ρj /(2κ) g′(zj ) qj zj zk ρj ρk /(8κ), t t | t − t | | t | | t − t | jY=1 1≤j Consider critical percolation in a bounded planar domain D containing the origin where ∂D is a simple closed curve. Let z1,...,z4 be four distinct points in clockwise order on ∂D. For every sufficiently small ε > 0, we may consider the triangular lattice of mesh ε with a vertex at the origin and the event = (D,ε,z1,z2,z3,z4) that the site at the origin is pivotal for an open crossing inA criticalA site percolation between two opposite arcs on ∂D determined by the four marked points in the union of the triangles of the grid that meet D. One could also consider the probability that the interfaces starting at an even number n of points on ∂D reach close to the origin. Essentially, the only difference is that the event has to be defined a bit differently, since there can be at most 3 distinct interfacesA that meet a hexagon. Observe that the event is equivalent to the event that the two (or n/2) percola- tion interfaces containingA the points zj’s all visit the boundary of the hexagon dual to the site at the origin. As the percolation interfaces are explored, the conditional probability of the event is a positive martingale that depends only on the domain cut by the explored segmentsA of the interfaces. This quantity should be (nearly) conformally invariant, modulo scaling by a power of the conformal radius of D with ′ j respect to the origin. In particular, the martingale will not depend on gt(z ) for the j boundary points zt . Referring to (9), we see that when κ = 6 [Smi01], by selecting ′ j each ρj = 2 the gt(z ) terms drop out, so that the martingale Mt simplifies to 2 M = g′(0)(n −1)/12 zj zk 1/3, t t | t − t | 1≤j 2 (10) (const+ o(1)) (εf ′(0))(n −1)/12 f(zj) f(zk) 1/3. × | − | 1≤j 2 M = g′(0)(n −1)/4 g′(zj ) zj zk . t t t | t − t | 1≤Yj 2 (11) (const+ o(1)) (εf ′(0))(n −1)/4 εf ′(zj ) f(zj) f(zk) . × | − | 1≤Yj≤n 1≤j The exponent of (n2 1)/4 agrees with the value computed in [Dup87, Ken00]. − Acknowledgments: We are grateful to Scott Sheffield for numerous conversations. 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