Stochastic partial differential equations for superprocesses in random environments

Let X=(Xₜ, t≥ 0) be a superprocess in a random environment described by a Gaussian noise Wᵍ=ᵍ(t,x), t≥ 0, x∈ Rᵈ\ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, Xₜ admits a jointly continuous density function Xₜ(x) that is a unique in law solution to a stochastic partial differential equation {align*} ∂ /∂ tX_t(x)=Δ/2 X_t(x)+√X_t(x) V̇(t,x)+X_t(x)Ẇ^g(t, x) , X_t(x)≥ 0, {align*} where V=(t,x), t≥ 0, x∈ R\ is a space-time white noise and is orthogonal with Wᵍ. When d≥ 2, we prove that Xₜ is singular and hence density does not exist.