Branching Brownian motion in strip: survival near criticality

TL;DR

This paper analyzes the survival probability of a branching Brownian motion in a shrinking interval near the critical width, revealing asymptotic behavior and a quasi-stationary limit as survival becomes unlikely.

Contribution

It provides exact asymptotics for near-critical survival probability and demonstrates the backbone thinning to a spine near criticality using spine techniques.

Findings

Exact asymptotics for near-critical survival probability

Existence of a quasi-stationary limit conditioned on survival

Backbone thins down to a spine as criticality is approached

Abstract

We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0,K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality. This paper is motivated by recent work on survival of near critical branching Brownian motion with absorption at the origin by Aidekon and Harris as well as the work of Berestycki, Berestycki and Schweinsberg.