Some quenched and annealed limit theorems of superprocesses in random environments

Let X=(Xₜ, t≥ 0) be a superprocess in a random environment described by a Gaussian noise W=(t,x), t≥ 0, x∈ Rᵈ\ white in time and colored in space with correlation kernel $g(x,y)$. When d≥ 3, under the condition that the correlation function $g(x,y)$ is bounded above by some appropriate function ḡ(x-y), we present the quenched and annealed Strong Law of Large Numbers and the Central Limit Theorems regarding the weighted occupation measure ∫₀ᵗ Xₛ ds as t→ ∞.