Stochastic partial differential equations for superprocesses in random environments
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913257547104256 |
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| author | Hong, Jieliang Xiong, Jie |
| author_facet | Hong, Jieliang Xiong, Jie |
| contents | 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) , \quad X_t(x)\geq 0, \end{align*} where $V=\{V(t,x), t\geq 0, x\in \mathbb{R}\}$ is a space-time white noise and is orthogonal with $W^g$. When $d\geq 2$, we prove that $X_t$ is singular and hence density does not exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic partial differential equations for superprocesses in random environments Hong, Jieliang Xiong, Jie Probability 60H15, 60G57, 60J80 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) , \quad X_t(x)\geq 0, \end{align*} where $V=\{V(t,x), t\geq 0, x\in \mathbb{R}\}$ is a space-time white noise and is orthogonal with $W^g$. When $d\geq 2$, we prove that $X_t$ is singular and hence density does not exist. |
| title | Stochastic partial differential equations for superprocesses in random environments |
| topic | Probability 60H15, 60G57, 60J80 |
| url | https://arxiv.org/abs/2403.03638 |