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 Electronic Communications in Probability > Vol. 15(2010) > Paper 3 open journal systems 


Almost sure finiteness for the total occupation time of an (d,α,β)-superprocess

Xiaowen Zhou, Concordia University


Abstract
For $0<α≤ 2$ and $0<β≤ 1$ let $X$ be the $(d,α,β)$-superprocess, i.e. the superprocess with $α$-stable spatial movement in $bR^d$ and $(1+β)$-stable branching. Given that the initial measure $X_0$ is Lebesgue on $bR^d$, Iscoe conjectured in [7] that the total occupational time $int_0^infty X_t(B)dt$ is a.s. finite if and only if $dβ<α$, where $B$ denotes any bounded Borel set in $reels^d$ 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 $2dβ<α$ and, on the other hand, $int_0^infty X_t(B)dt=infty$ a.s. if $dβ> α$.

For $2dβ<α$, 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 $(d,al,β)$-branching particle system with Lebesgue initial intensity measure.


Full text: PDF

Pages: 22-31

Published on: February 10, 2010


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Electronic Communications in Probability. ISSN: 1083-589X