| Título: | Almost sure finiteness for the total occupation time of an $(d,\alpha,\beta)$-superprocess |
| Autores: | Zhou, Xiaowen; Concordia University |
| Fecha: | 2010-01-01 |
| Publicador: | Electronic communications in probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: | No aplica |
| Descripción: | 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. |
| Idioma: | No aplica |
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