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 $|x-y|^{-\alpha}$, $0\leq \alpha\leq 2$, 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 $g$.
Highlights
A system of branching particles whose branching probabilities depend on the random environment the particles are in is studied by Mytnik [20]
Let ∆ be the d-dimensional Laplacian operator. It was proved in [20] that the high-density limit of the above system converges to a measure-valued process X which is a solution to the following martingale problem (MP): ∀ φ ∈ Cb2(Rd), t
The aim of this paper is to study the local extinction of this process X
Summary
A system of branching particles whose branching probabilities depend on the random environment the particles are in is studied by Mytnik [20]. Note that the equation (1.3) was studied by Dawson and Salehi [6] in the case of homogeneous noise W , that is, g(x, y) = q(x − y) for some function q They proved (see Theorem 3.4 and Remark 4.2 in [6]) that if q(0) is sufficiently small, there exists a non-trivial limiting longterm distribution of a solution to (1.3). This and the fact that superBrownian motion starting at the Lebesgue measure persists in dimensions d ≥ 3 allows us to make the following conjecture.
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