论文标题
在关键的超级布朗尼运动的空球上
On the empty balls of a critical super-Brownian motion
论文作者
论文摘要
令$ \ {x_t \} _ {t \ geq0} $为$ d $ - 二维关键的批判性超棕色运动，始于泊松随机度量，其强度是lebesgue度量。用$ r_t表示：= \ sup \ {u> 0：x_t（\ {x \ in \ mathbb {r}^d：| x | <u \}）= 0 \} $最大的空球的半径以$ x_t $为中心。在这项工作中，我们证明，对于$ r> 0 $，$$ \ lim_ {t \ to \ infty} \ mathbb {p} \ left（\ frac {r_t} {t^{（1/d）\ wedge（1/d）\ wedge（wedge（3-d）满足$ \ lim_ {r \ to \ infty} \ frac {a_d（r）} {r^{| d-2 |+d-2 |+d \ d \ ind _ {\ {\ {\ {d = 2 \}}}}} = c $ for（0，\ infty）$仅在$ d $上。
Let $\{X_t\}_{t\geq0}$ be a $d$-dimensional critical super-Brownian motion started from a Poisson random measure whose intensity is the Lebesgue measure. Denote by $R_t:=\sup\{u>0: X_t(\{x\in\mathbb{R}^d:|x|< u\})=0\}$ the radius of the largest empty ball centered at the origin of $X_t$. In this work, we prove that for $r>0$, $$\lim_{t\to\infty}\mathbb{P}\left(\frac{R_t}{t^{(1/d)\wedge(3-d)^+}}\geq r\right)=e^{-A_d(r)},$$ where $A_d(r)$ satisfies $\lim_{r\to\infty}\frac{A_d(r)}{r^{|d-2|+d\ind_{\{d=2\}}}}=C$ for some $C\in(0,\infty)$ depending only on $d$.
