On the empty balls of a critical super-Brownian motion

TL;DR

This paper investigates the asymptotic behavior of the largest empty ball in a critical super-Brownian motion in , establishing a limit law for its scaled radius and characterizing the growth of the associated rate function.

Contribution

It provides a precise asymptotic distribution for the largest empty ball in critical super-Brownian motion, a novel result in the spatial structure analysis of such processes.

Findings

The scaled radius of the largest empty ball converges in distribution to an exponential form.

The rate function $A_d(r)$ exhibits polynomial growth with degree depending on the dimension.

The asymptotic behavior depends critically on the dimension $d$, with different regimes for $d=2$ and $d eq2$.

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 $d$.