Exit Laws and Excessive Functions for Superprocesses

Abstract

Let $\xi$ be a Markov process with transition function $p(r,x;\,t,\,dy)$ and let X be the corresponding Dawson--Watanabe superprocess (i.e., the superprocess with the branching characteristic $\psi(u)=\gamma u^2$). Denote by $\cal P$ the transition function of X and put $$ p_n(r,x;\,t,\,dy)=\prod_{i=1}^n p(r,x_i;\,t,\,dy_i). $$ To every $p_n$-exit law $\ell$ there corresponds a $\cal P$-exit law $L_\ell$ such that, for every t, $L_\ell^t(\mu)$ is a polynomial of degree n in $\mu$ with the leading term $\langle \ell^t,\mu^n\rangle $. Every polynomial $\cal P$-exit law has a unique representation of the form $L_{\ell_1}+\cdots+L_{\ell_n}$, where $\ell_k$ is a $p_k$-exit law.