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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.20094 |
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| _version_ | 1866911613992304640 |
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| author | Chen, Zhen-Qing Ren, Yan-Xia Zhao, Guohuan |
| author_facet | Chen, Zhen-Qing Ren, Yan-Xia Zhao, Guohuan |
| contents | We consider a super-Brownian motion $\{X_t, t\geq 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\geq 0, x\in\mathbb{R}^d\}$ whose correlation function is given by $\mathcal{C} (x,y)(t \wedge s)$. The process takes values in $\mathcal{M}(\mathbb{R}^d)$, the space of Radon measures on $\mathbb{R}^d$. It can be characterized through a conditional Laplace transform by a parabolic stochastic partial differential equation driven by $W(t, x)$. Suppose that $\mathcal{C} (x, y)\leq g(x-y)$ for some bounded positive function $g$ on $\mathbb{R}^d$ and the initial distribution of process $X$ is the Lebesgue measure $m$ on $\mathbb{R}^d$. We prove that for dimension $d\geq 3$, whenever $$
\sup_{x\in \mathbb{R}^d} \int_{\mathbb{R}^d} |x-y|^{2-d} g(y)dy< \frac{8 (d-2) π^{d/2}}{d 2^d Γ\left(d/2-1\right)}, $$ the distribution of $X_t$ converges weakly as $t \to \infty$ to a non-trivial invariant probability distribution $π^m$ on $\mathcal{M}(\mathbb{R}^d)$ with mean measure $m$. This result in particular gives an affirmative answer to Conjecture 1.4 of Mytnik and Xiong (Electron. J. Probab. 12: 1349-1378 (2007)). We further show that given $ Θ\in C^β(\mathbb{R}^d)$ $(β>1)$, when $\mathcal{C}(x,y)= a Θ(x-y)$ with $a$ being large enough, the superprocess $X$ suffers local extinction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Persistence and local extinction for superprocesses in random environments Chen, Zhen-Qing Ren, Yan-Xia Zhao, Guohuan Probability We consider a super-Brownian motion $\{X_t, t\geq 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\geq 0, x\in\mathbb{R}^d\}$ whose correlation function is given by $\mathcal{C} (x,y)(t \wedge s)$. The process takes values in $\mathcal{M}(\mathbb{R}^d)$, the space of Radon measures on $\mathbb{R}^d$. It can be characterized through a conditional Laplace transform by a parabolic stochastic partial differential equation driven by $W(t, x)$. Suppose that $\mathcal{C} (x, y)\leq g(x-y)$ for some bounded positive function $g$ on $\mathbb{R}^d$ and the initial distribution of process $X$ is the Lebesgue measure $m$ on $\mathbb{R}^d$. We prove that for dimension $d\geq 3$, whenever $$ \sup_{x\in \mathbb{R}^d} \int_{\mathbb{R}^d} |x-y|^{2-d} g(y)dy< \frac{8 (d-2) π^{d/2}}{d 2^d Γ\left(d/2-1\right)}, $$ the distribution of $X_t$ converges weakly as $t \to \infty$ to a non-trivial invariant probability distribution $π^m$ on $\mathcal{M}(\mathbb{R}^d)$ with mean measure $m$. This result in particular gives an affirmative answer to Conjecture 1.4 of Mytnik and Xiong (Electron. J. Probab. 12: 1349-1378 (2007)). We further show that given $ Θ\in C^β(\mathbb{R}^d)$ $(β>1)$, when $\mathcal{C}(x,y)= a Θ(x-y)$ with $a$ being large enough, the superprocess $X$ suffers local extinction. |
| title | Persistence and local extinction for superprocesses in random environments |
| topic | Probability |
| url | https://arxiv.org/abs/2604.20094 |