An ergodic theorem for the maximum of branching Brownian motion with absorption

TL;DR

This paper proves that the maximum of branching Brownian motion with absorption converges almost surely to a randomly shifted Gumbel distribution, revealing new probabilistic behavior in such processes.

Contribution

It establishes an ergodic theorem for the maximum of absorbed branching Brownian motion, a novel result in the study of stochastic processes with absorption.

Findings

Empirical distribution of the maximum converges almost surely.

The limiting distribution is a randomly shifted Gumbel distribution.

Provides new insights into the extremal behavior of absorbed branching processes.

Abstract

In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions and are killed upon hitting the absorption barrier. We prove that the empirical distribution function of the maximum of this process converges almost surely to a randomly shifted Gumbel distribution.