Occupation times for superprocesses in random environments
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909890675474432 |
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| author | Cheng, Ziling Hong, Jieliang Yao, Dan |
| author_facet | Cheng, Ziling Hong, Jieliang Yao, Dan |
| contents | Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t, x),t\geq 0,x\in\mathbb{R}^d\}$ 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 $d\leq 3$, whereas it is singular with respect to Lebesgue measure in $d\geq 4$. Regarding the absolutely continuous case in $d\leq 3$, 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 $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Occupation times for superprocesses in random environments Cheng, Ziling Hong, Jieliang Yao, Dan Probability Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment governed by a Gaussian noise $W=\{W(t, x),t\geq 0,x\in\mathbb{R}^d\}$ 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 $d\leq 3$, whereas it is singular with respect to Lebesgue measure in $d\geq 4$. Regarding the absolutely continuous case in $d\leq 3$, 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 $t$. |
| title | Occupation times for superprocesses in random environments |
| topic | Probability |
| url | https://arxiv.org/abs/2511.04535 |