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SLE coordinate changes. (English) Zbl 1094.82007
It is known [O. Schramm, Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093); W. Werner Lect. Notes Math. 1840, 109–195 (2004; Zbl 1057.60078)] that SLE (i.e. Stochastic Loewner Evolution) describes random paths in the plane by specifying a stochastic differential equation satisfied by the conformal maps in the complement, and comes in several different variations, as chordal, radial and dipolar forms.
Let $$X\in\{\mathbb{D},\mathbb{H}\}$$, here $$\mathbb{D}$$ and $$\mathbb{H}$$ are the unit circle and half plane in the complex plane, respectively. Let $$\Psi_X(w,z)$$ denote the corresponding Loewner vector field, i.e. $$\Psi_{\mathbb{D}}(w,z)=-z\frac{z+w}{z-w}$$, $$\Psi_{\mathbb{H}}(w,z)=\frac{2}{z-w}$$. In the paper SLE is defined with a force point in the interior domain.
Let $$\kappa\geq 0$$, $$m\in\mathbb{N}$$, $$\rho_1,\dots,\rho_m\in\mathbb{R}$$. Let $$V^1,\dots,V^m\in\overline{X}$$ (here $$\overline{X}$$ is the closure of $$X$$), and $$w_0\in\partial\setminus \{\infty,V^1,\dots,V^m\}$$ and $$B_t$$ be standard one-dimensional Browian motion. Consider the solution of the system $dW_t=G_X(W_t,dB_t,dt)+\sum_{j=1}^m\frac{\rho_j}{2}\widetilde\Psi_X(V^j,W_t)\,dt,$
$dV^j_t=\Psi_X(W_t,V_t^j)\,dt, \quad j=1,\dots,m$ starting at $$W_0=w_0$$ and $$V_0^j=V^j$$,$$j=1,\dots,m$$ up to the first time $$\tau$$ such that for some $$j$$ $$\inf\{| W_t-V^j_t| : t<\tau\}=0$$, here $G_{\mathbb{D}}(W_t,dB_t,dt)=-(\kappa/2)W_tdt+i\sqrt{\kappa}W_tdB_t, \;\;G_{\mathbb{H}}(W_t,dB_t,dt)=-\sqrt{\kappa}dB_t,$
$\widetilde\Psi_X(z,w)=\frac{\Psi_X(z,w)+\Psi_X(I_X(z),w)}{2},$ where $$I_X$$ denotes the inversion in $$\partial X$$. Let $$g_t(z)$$ be the solution of $$\partial_tg_t(z)=\Psi_X(W_t,g_t(z))$$ starting at $$g_0(z)=z$$. The evolution $$t\to K_t:=\{z\in\overline{X}: t_z\leq t\}$$ is called $$X\text{-SLE}(\kappa;\rho_1,\dots,\rho_m)$$.
The main result of the paper is following
Theorem Let $$X,Y\in \{\mathbb{D},\mathbb{H}\}$$, and $$\psi:X\to Y$$ be a Möbius transformation such that $$\psi(X)=Y$$. Let $$w_0\in\partial\setminus \{\infty\}$$, $$V^1,\dots,V^m\in\overline{X}\setminus\{w_0\}$$. Suppose that $$\rho_1,\dots,\rho_m\in\mathbb{R}$$ satisfy $$\sum_{j=1}^m\rho_j=\kappa-6$$. Then the image under $$\psi$$ of the $$X-\text{SLE}(\kappa;\rho_1,\dots,\rho_m)$$ starting from $$(w_0,V^1,\dots,V^m)$$ and stopped at some a.s. positive time has the same law as a time change of the $$Y-\text{SLE}(\kappa;\rho_1,\dots,\rho_m)$$ starting from $$(\psi(w_0),\psi(V^1),\dots,\psi(V^m))$$ and stopped at an a.s. positive time.
The authors also discuss the martingale describing the Radon-Nikodym derivative of starting SLE$$(\kappa;\rho_1,\dots,\rho_m)$$ with respect to SLE$$(\kappa)$$.

##### MSC:
 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics 60G48 Generalizations of martingales 60D05 Geometric probability and stochastic geometry
##### Keywords:
Stochastic Loewner evolution
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