On properties of a class of strong limits for supercritical superprocesses

Abstract

Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_μ\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $ϕ_0$ be a positive eigenfunction corresponding to the first eigenvalue $λ_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-λ_0t}\langleϕ_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. When the $L\log L$ condition may not be satisfied, we recently proved in (arXiv:1708.04422) that there exist a non-negative function $γ_t$ on $[0, \infty)$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $μ$ on $E$, $$ \lim_{t\to\infty}γ_t\langle ϕ_0,X_t\rangle =W,\qquad\mbox{a.s.-}\mathbb{P}_μ. $$ In this paper, we mainly investigate properties of $W$. We prove that $W$ has strictly positive density on $(0,\infty)$. We also investigate the small value probability and tail probability problems of $W$.