Persistence and local extinction for superprocesses in random environments

We consider a super-Brownian motion $\{X_t, t\geq 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\geq 0, x\in\mathbb{R}^d\}$ 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 $\mathcal{C} (x, y)\leq g(x-y)$ 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 $d\geq 3$, whenever $$\sup_{x\in \mathbb{R}^d} \int_{\mathbb{R}^d} |x-y|^{2-d} g(y)dy< \frac{8 (d-2) \pi^{d/2}}{d 2^d \Gamma \left(d/2-1\right)},$$ the distribution of $X_t$ converges weakly as $t \to \infty$ to a non-trivial invariant probability distribution $\pi^m$ 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 $\Theta \in C^\beta(\mathbb{R}^d)$ $(\beta>1)$, when $\mathcal{C}(x,y)= a \Theta (x-y)$ with $a$ being large enough, the superprocess $X$ suffers local extinction.