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

TL;DR

This paper investigates the lower large deviation probabilities of the supremum of the support of super-Brownian motion, complementing previous work on its asymptotic distribution and upper deviations.

Contribution

It provides the first analysis of the lower large deviation behavior of the supremum of super-Brownian motion support under survival conditioning.

Findings

Derived asymptotics for the probability that the supremum is significantly below its typical growth rate.

Extended understanding of the support's extremal behavior in super-Brownian motion.

Complemented previous results on upper deviations and distributional limits.

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 $m(t)=c_0t-c_1\log t$. In the same paper, we also studied the upper large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t>\delta c_0t) $ for $\delta\ge 1$. In this paper, we study the lower large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t\le \delta c_0t|\mathcal{S}) $ for $\delta<1$, where $\mathcal{S}$ is the survival event.