On properties of a class of strong limits for supercritical superprocesses

Abstract

Suppose that $X=\{X_t,t\ge 0;{\mathbb{P}}_{\hat{I}}¼\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\ddot{I}{•}_0$ be a positive eigenfunction corresponding to the first eigenvalue $\hat{I}{»}_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\hat{I}{»}_{0}t}⟨\ddot{I}{•}_0,X_t⟩$ 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 ${\hat{I}}_t^3$ on $[0,\infty )$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $\hat{I}¼$ 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 $W$.

Minor typos are corrected. The Chinese version will appear in Sci. Sin. Math