Stochastic partial differential equations for superprocesses in random environments

2024/03/06 by Jieliang Hong, Jie Xiong, Hong, Jieliang +1
Decision Sciences · Mathematics · #60G57 #60H15 #60J80 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Simulation Techniques and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2403.03638

openalex publication_date 2024/03/06 · openalex created_date 2024/03/08 · openalex updated_date 2026/07/28

Abstract

Let X=(X_t, t≥ 0) be a superprocess in a random environment described by a Gaussian noise W^g=\W^g(t,x), t≥ 0, x∈ 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 (∂ )/(∂ t)X_t(x)=\fracΔ2 X_t(x)+√(X_t(x)) V(t,x)+X_t(x)W^g(t, x) , X_t(x)≥ 0, where V=\V(t,x), t≥ 0, x∈ ℝ\ is a space-time white noise and is orthogonal with W^g. When d≥ 2, we prove that X_t is singular and hence density does not exist.

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