Occupation times for superprocesses in random environments

2025/11/06 by Ziling Cheng, Cheng, Ziling, Jieliang Hong +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2511.04535

openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28

Abstract

Let X=(X_t, t≥ 0) be a superprocess in a random environment governed by a Gaussian noise W=\W(t, x),t≥ 0,x∈ℝ^d\ white in time and colored in space with correlation kernel g. We consider the occupation time process of the model starting from a finite measure. It is shown that the occupation time process of X is absolutely continuous with respect to Lebesgue measure in d≤ 3, whereas it is singular with respect to Lebesgue measure in d≥ 4. Regarding the absolutely continuous case in d≤ 3, we further prove that the associated density function is jointly Hölder continuous based on the Tanaka formula and moment formulas, and derive the Hölder exponents with respect to the spatial variable x and the time variable t.

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