Journal Article

A large deviation theorem for a supercritical super-Brownian motion with absorption

Zhu, Yaping

2023

Overview

We consider a one-dimensional superprocess with a supercritical local branching mechanism $\\psi$ , where particles move as a Brownian motion with drift $-\\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\\rho<\\sqrt{2\\alpha}$ , where $\\alpha=-\\psi'(0)$ . When $\\rho<\\sqrt{2 \\alpha}$ , Kyprianou et al. [18] proved that $\\lim_{t\\to \\infty}R_t/t =\\sqrt{2\\alpha}-\\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\\mathbb{P}_{\\delta_x} (R_t >\\gamma t+\\theta)$ as $t \\to \\infty$ , where $\\gamma >\\sqrt{2 \\alpha} -\\rho$ , $\\theta \\ge 0$ . As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].

• Publisher: Cambridge University Press
• Subject: Brownian motion / Deviation / Markov analysis / Original Article / Partial differential equations / Probability distribution / Theorems