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2007 Local extinction for superprocesses in random environments
Leonid Mytnik, Jie Xiong
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Electron. J. Probab. 12: 1349-1378 (2007). DOI: 10.1214/EJP.v12-457


We consider a superprocess in a random environment represented by a random measure which is white in time and colored in space with correlation kernel $g(x,y)$. Suppose that $g(x,y)$ decays at a rate of $|x-y|^{-\alpha}$, $0\leq \alpha\leq 2$, as $|x-y|\to\infty$. We show that the process, starting from Lebesgue measure, suffers longterm local extinction. If $\alpha < 2$, then it even suffers finite time local extinction. This property is in contrast with the classical super-Brownian motion which has a non-trivial limit when the spatial dimension is higher than 2. We also show in this paper that in dimensions $d=1,2$ superprocess in random environment suffers local extinction for any bounded function $g$.


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Leonid Mytnik. Jie Xiong. "Local extinction for superprocesses in random environments." Electron. J. Probab. 12 1349 - 1378, 2007.


Accepted: 3 November 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1130.60095
MathSciNet: MR2354161
Digital Object Identifier: 10.1214/EJP.v12-457

Vol.12 • 2007
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