Persistence and local extinction for superprocesses in random environments

Abstract

We consider a super-Brownian motion $\{X_t,t\ge 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\ge 0,x\in {\mathbb{R}}^d\}$ whose correlation function is given by $\mathcal{C}(x,y)(t\wedge s)$. The process takes values in $\mathcal{M}({\mathbb{R}}^d)$, the space of Radon measures on ${\mathbb{R}}^d$. It can be characterized through a conditional Laplace transform by a parabolic stochastic partial differential equation driven by $W(t,x)$. Suppose that $\mathcal{C}(x,y)\le g(x-y)$ for some bounded positive function $g$ on ${\mathbb{R}}^d$ and the initial distribution of process $X$ is the Lebesgue measure $m$ on ${\mathbb{R}}^d$. We prove that for dimension $d\ge 3$, whenever $$\underset{x\in {\mathbb{R}}^d}{\sup }{\int }_{{\mathbb{R}}^d}|x-y{|}^{2-d}g(y)dy<\frac{8(d-2)\ddot{I}{€}^{d/2}}{d{2}^d\hat{I}“\left(d/2-1\right)},$$ the distribution of $X_t$ converges weakly as $t\to \infty$ to a non-trivial invariant probability distribution $\ddot{I}{€}^m$ on $\mathcal{M}({\mathbb{R}}^d)$ with mean measure $m$. This result in particular gives an affirmative answer to Conjecture 1.4 of Mytnik and Xiong (Electron. J. Probab. 12: 1349-1378 (2007)). We further show that given $\hat{I}˜\in C^2({\mathbb{R}}^d)$ $({\hat{I}}^2>1)$, when $\mathcal{C}(x,y)=a\hat{I}˜(x-y)$ with $a$ being large enough, the superprocess $X$ suffers local extinction.

37 pages