On the empty balls of a critical super-Brownian motion

2022/04/25 by Jie Xiong, Xiong, Jie, Shuxiong Zhang +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2204.11468

openalex publication_date 2022/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \X_t\_t≥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∈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→∞ℙ(\fracR_tt^{(1/d)\wedge(3-d)+}≥ r)=e^-A_d(r), where A_d(r) satisfies lim_r→∞\fracA_d(r)r^|d-2|+d\ind_\d=2\=C for some C∈(0,∞) depending only on d.

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