Large deviations for the branching Brownian motion in presence of selection or coalescence

TL;DR

This paper investigates large deviation functions for the position of the rightmost particle in generalized branching Brownian motions with selection or coalescence, providing upper bounds and revealing non-analytic parameter dependencies.

Contribution

It introduces methods to estimate upper bounds for large deviation functions in generalized BBM models with selection or coalescence, highlighting non-analytic behaviors.

Findings

Upper bounds for large deviation functions obtained

Non-analytic dependence on parameters identified

Applicable to L-BBM, N-BBM, and CBRW models

Abstract

The large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk (BRW), or in a branching Brownian motion (BBM). More recently a number of generalizations of the BBM and of the BRW have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles. Here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations: the $L$-BBM, the $N$-BBM, and the CBRW (coalescing branching random walk) which is closely related to the noisy FKPP equation. Our approach allows us to obtain only upper bounds on these large deviation functions. One noticeable feature of our results is their non analytic dependence on the parameters (such as the coalescence rate in the CBRW).