Upper deviation probabilities for the range of a supercritical super-Brownian motion

2024/05/30 by Shuxiong Zhang, Zhang, Shuxiong · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2405.19756

openalex publication_date 2024/05/30 · openalex created_date 2024/06/01 · openalex updated_date 2026/07/28

Abstract

Let \X_t\_{t≥ 0} be a d-dimensional supercritical super-Brownian motion started from the origin with branching mechanism ψ. Denote by R_t:=inf\r>0:X_s(\x∈ ℝ^d:|x|≥ r\)=0,~∀~0≤ s≤ t\ the radius of the minimal ball (centered at the origin) containing the range of \X_s\_{s≥ 0} up to time t. In \citePinsky, Pinsky proved that condition on non-extinction, lim_t→∞R_t/t=√(2β) in probability, where β:=-ψ'(0). Afterwards, Engländer \citeEnglander04 studied the lower deviation probabilities of R_t. For the upper deviation probabilities, he \cite[Conjecture 8]Englander04 conjectured that for ρ>√ 2β, lim_t→∞(1)/(t)logℙ(R_t≥ ρt)=-((ρ²)/(2)-β). In this note, we confirmed this conjecture.

Cited by

• On the maximal displacement of subcritical branching random walks with or without killing

Related

• Lower deviation for the supremum of the support of super-Brownian motion
• Large deviation probabilities for the range of a d-dimensional supercritical branching random walk
• Upper deviation probabilities for level sets of a supercritical branching random walk
• The extremal process of super-Brownian motion
• Limiting distributions for a class of super-Brownian motions with spatially dependent branching mechanisms