A note 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 $r>0$,

$$\underset{t\to \mathrm{\infty }}{lim}\mathbb{P}\left(\frac{R_t}{t^{(1∕d)\wedge {(3-d)}^{+}}}\ge r\right)=e^{-A_d(r)},$$

where $A_d(r)$ satisfies

$$\underset{r\to \mathrm{\infty }}{lim}\frac{A_d(r)}{r^{|d-2|+d{\mathbf{1}}_{\{d=2\}}}}=C$$

for some $C\in (0,\mathrm{\infty })$ depending only on d.