An exponential estimate for the extinction time of the branching random walk on a cube

TL;DR

This paper establishes an exponential bound on the probability that a supercritical branching random walk on a cube survives beyond a certain time, applicable to both discrete and continuous spaces.

Contribution

It provides the first exponential estimate for the extinction time of supercritical BRWs on a cube, covering both discrete and continuous models.

Findings

Proves exponential decay of survival probability for BRWs

Applies to both discrete and continuous space models

Provides bounds useful for understanding BRW extinction dynamics

Abstract

We prove the exponential estimate \begin{equation*} P \{ s < \tau < \infty \} \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C, q >0$ are constants and $ \tau $ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs.