Some quenched and annealed limit theorems of superprocesses in random environments

TL;DR

This paper establishes quenched and annealed limit theorems, including strong laws and central limit theorems, for superprocesses in a Gaussian random environment with spatial correlation, in dimensions three and higher.

Contribution

It provides the first rigorous derivation of quenched and annealed limit theorems for superprocesses in correlated Gaussian environments in high dimensions.

Findings

Proved quenched and annealed strong laws of large numbers.

Established central limit theorems for the occupation measure.

Extended results to environments with bounded correlation functions.

Abstract

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W=\{W(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. When $d\geq 3$, under the condition that the correlation function $g(x,y)$ is bounded above by some appropriate function $\bar{g}(x-y)$, we present the quenched and annealed Strong Law of Large Numbers and the Central Limit Theorems regarding the weighted occupation measure $\int_0^t X_s ds$ as $t\to \infty$.