Local extinction for superprocesses in random environments
Leonid Mytnik, Technion
Jie Xiong, University of Tennessee and Hebei Normal University
Abstract
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|-α, 0≤α≤ 2, as |x-y|->∞. We
show that the process, starting from Lebesgue measure, suffers
longterm local extinction. If α<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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