On the empty balls of a critical super-Brownian motion

Abstract: 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 $d$.