Some quenched and annealed limit theorems of superprocesses in random environments

Let $X=(X_t,t\ge 0)$ be a superprocess in a random environment described by a Gaussian noise $W=\{W(t,x),t\ge 0,x\in {\mathbb{R}}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. When $d\ge 3$, 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 $t\to \mathrm{\infty }$.