Upper deviation probabilities for the range of a supercritical super-Brownian motion
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929366156443648 |
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| author | Zhang, Shuxiong |
| author_facet | Zhang, Shuxiong |
| contents | Let $\{X_t\}_{t\geq 0 }$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $ψ$. Denote by $R_t:=\inf\{r>0:X_s(\{x\in \mathbb{R}^d:|x|\geq r\})=0,~\forall~0\leq s\leq t\}$ the radius of the minimal ball (centered at the origin) containing the range of $\{X_s\}_{s\geq 0 }$ up to time $t$. In \cite{Pinsky}, Pinsky proved that condition on non-extinction, $\lim_{t\to\infty}R_t/t=\sqrt{2β}$ in probability, where $β:=-ψ'(0)$. Afterwards, Engländer \cite{Englander04} studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, he \cite[Conjecture 8]{Englander04} conjectured that for $ρ>\sqrt {2β}$,
$$ \lim_{t\to\infty}\frac{1}{t}\log\mathbb{P}(R_t\geq ρt)=-\left(\frac{ρ^2}{2}-β\right). $$ In this note, we confirmed this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper deviation probabilities for the range of a supercritical super-Brownian motion Zhang, Shuxiong Probability Let $\{X_t\}_{t\geq 0 }$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $ψ$. Denote by $R_t:=\inf\{r>0:X_s(\{x\in \mathbb{R}^d:|x|\geq r\})=0,~\forall~0\leq s\leq t\}$ the radius of the minimal ball (centered at the origin) containing the range of $\{X_s\}_{s\geq 0 }$ up to time $t$. In \cite{Pinsky}, Pinsky proved that condition on non-extinction, $\lim_{t\to\infty}R_t/t=\sqrt{2β}$ in probability, where $β:=-ψ'(0)$. Afterwards, Engländer \cite{Englander04} studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, he \cite[Conjecture 8]{Englander04} conjectured that for $ρ>\sqrt {2β}$, $$ \lim_{t\to\infty}\frac{1}{t}\log\mathbb{P}(R_t\geq ρt)=-\left(\frac{ρ^2}{2}-β\right). $$ In this note, we confirmed this conjecture. |
| title | Upper deviation probabilities for the range of a supercritical super-Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2405.19756 |