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Stochastic partial differential equations for superprocesses in random environments
Jieliang Hong, Jie Xiong
https://arxiv.org/abs/2403.03638 https://

Stochastic partial differential equations for superprocesses in random environments
Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W^g=\{W^g(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, $X_t$ admits a jointly continuous density function $X_t(x)$ that is a unique in law solution to a stochastic partial differential equation \begin{align*} \frac{\partial }{\partial t}X_t(x)=\fracΔ{2} X_t(x)+\sqrt{X_t(x)} \dot{V}(t,x)+X_t(x)\dot{W}^g(t, x) , \qu…