On the empty balls of a critical super-Brownian motion

Abstract

Let $\{X_t{\}}_{t\ge 0}$ 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 $r>0$, $$\underset{t\to \infty }{\lim }\mathbb{P}\left(\frac{R_t}{t^{(1/d)\wedge (3-d{)}^{+}}}\ge r\right)=e^{-A_d(r)},$$ 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 $d$.