Exit densities of Super--Brownian motion as extreme X-harmonic functions

Abstract

Let $X$ be a super-Brownian motion (SBM) defined on a domain $E\subset R^n$ and $(X_D)$ be its exit measures indexed by sub-domains of $E$. The relationship between the equation $1/2 \Delta u=2 u^2$ and Super-Brownian motion (SBM) is analogous to the relationship between Brownian motion and the Laplace's equation, and substantial progress has been made on the study of the solutions of this semi-linear p.d.e. exploring this analogy. An area that remains to be explored is Martin boundary theory. Martin boundary in the semi-linear case is defined as the convex set of extreme $X$-harmonic functions which are functions on the space of finite measures supported in a domain $E$ of $R^d$ and characterized by a mean value property with respect to the Super-Brownian law. So far no probabilistic construction of Martin boundary is known. In this paper, we consider a bounded smooth domain $D$, and we investigate exit densities of SBM, a certain family of $X$ harmonic functions, $H^{\nu}$, indexed by finite measures $\nu$ on $\partial{D}$, These densities were first introduced by E.B. Dynkin and also identified by T.Salisbury and D. Sezer as the extended X-harmonic functions corresponding to conditioning SBM on its exit measure $X_D$ being equal to $\nu$. $H^{\nu}(\mu)$ can be thought as the analogue of the Poisson kernel for Brownian motion. It is well known that Poisson kernel for a smooth domain $D$ is equivalent to the so called Martin kernel, the class of extreme harmonic functions for $D$. We show that a similar result is true for Super-Brownian motion as well, that is $H^{\nu}$ is extreme for almost all $\nu$ with respect to a certain measure.