Lower deviation for the supremum of the support of super-Brownian motion

2022/12/08 by Ren, Yan-Xia, Song, Renming, Zhang, Rui
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2212.04178

Abstract

We study the asymptotic behavior of the supremum M_t of the support of a supercritical super-Brownian motion. In our recent paper (Stoch. Proc. Appl. 137 (2021), 1-34), we showed that, under some conditions, M_t-m(t) converges in distribution to a randomly shifted Gumbel random variable, where m(t)=c₀t-c₁log t. In the same paper, we also studied the upper large deviation of M_t, i.e., the asymptotic behavior of ℙ(M_t>δc₀t) for δ≥ 1. In this paper, we study the lower large deviation of M_t, i.e., the asymptotic behavior of ℙ(M_t≤ δc₀t|S) for δ<1, where S is the survival event.

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