zbMATH — the first resource for mathematics

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\((\kappa)\).

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
60G48 Generalizations of martingales
60D05 Geometric probability and stochastic geometry
Full Text: EuDML arXiv