Almost sure finiteness for the total occupation time of an $(d,\alpha,\beta)$-superprocess

Abstract

For $0<\alpha\leq 2$ and $0<\beta\leq 1$ let $X$ be the $(d,\alpha,\beta)$-superprocess, i.e. the superprocess with $\alpha$-stable spatial movement in $R^d$ and $(1+\beta)$-stable branching. Given that the initial measure $X_0$ is Lebesgue on $R^d$, Iscoe conjectured in [7] that the total occupational time $\int_0^\infty X_t(B)dt$ is a.s. finite if and only if $d\beta < \alpha$, where $B$ denotes any bounded Borel set in $R^d$ with non-empty interior.

In this note we give a partial answer to Iscoe's conjecture by showing that $\int_0^\infty X_t(B)dt<\infty$ a.s. if $2d\beta < \alpha$ and, on the other hand, $\int_0^\infty X_t(B)dt=\infty$ a.s. if $d\beta > \alpha$.

For $2d\beta< \alpha$, our result can also imply the a.s. finiteness of the total occupation time (over any bounded Borel set) and the a.s. local extinction for the empirical measure process of the $(d,\alpha,\beta)$-branching particle system with Lebesgue initial intensity measure.