Stochastic Komatu-Loewner evolutions and BMD domain constant

Abstract

Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $H$. Given a Jordan arc $\gamma\subset D$ starting at $\partial H$, let $g_t$ be the unique conformal map from $D\setminus\gamma[0,t]$ onto a standard slit domain $D_t={\mathbb H} \setminus \cup_{k=1}^N C_k(t)$ satisfying the hydrodynamic normalization at infinity. It has been established recently that $g_t$ satisfies an ODE called a Komatu-Loewner equation in terms of the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$. We randomize the Jordan arc $\gamma$ according to a system of probability measures on the family of equivalence classes of Jordan arcs that enjoy a domain Markov property and a certain conformal invariance property. We show that the induced process $(\xi(t), {\bf s}(t))$ satisfies a Markov type stochastic differential equation, where $\xi(t)$ is a motion on $\partial {\mathbb H}$ and ${\bf} s(t)$ represents the motion of the endpoints of the slits $\{C_k(t),\; 1\le k\le N \}.$ Conversely, given such functions $\alpha$ and $b$ with local Lipschitz continuity, the corresponding SDE admits a unique solution $(\xi(t), {\bf s}(t))$. The latter produces random conformal maps $g_t(z)$ via the Komatu-Loewner equation. The resulting family of random growing hulls $\{F_t\}$ from the conformal mappings is called ${\rm SKLE}_{\alpha,b}.$ We show that it enjoys a certain scaling property and a domain Markov property. Among other things, we further prove that ${\rm SKLE}_{\alpha,-b_{\rm BMD}}$ for a constant $\alpha >0$ has a locality property if and only if $\alpha = \sqrt{6}$, where $b_{\rm BMD}$ is a BMD-domain constant that describes the discrepancy of a standard slit domain from ${\mathbb H}$ relative to BMD.