Supercritical Superprocesses: Proper Normalization and Non-degenerate Strong Limit

Abstract

Suppose that $X=\{X_t,t\ge 0;{\mathbb{P}}_{\mu }\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let ${\varphi }_0$ be a positive eigenfunction corresponding to the first eigenvalue ${\lambda }_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-{\lambda }_{0}t}⟨{\varphi }_0,X_t⟩$ is a positive martingale. Let $M_{\infty }$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_{\infty }$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We prove that, under some conditions, there exist function ${\gamma }_t$ on $[0,\infty )$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $\mu$ on $E$, $$ \lim_{t\to\infty}γ_t\langle ϕ_0,X_t\rangle =W,\qquad\mbox{a.s.-}\mathbb{P}_μ. $$ We also give the almost sure limit of ${\gamma }_t⟨f,X_t⟩$ for a class of general test functions $f$.