Slowdown in branching Brownian motion with inhomogeneous variance

TL;DR

This paper refines the understanding of the maximal displacement in inhomogeneous branching Brownian motion, revealing precise asymptotics and a law of large numbers for the maximum's position.

Contribution

It provides a detailed asymptotic expansion for the maximum particle position in inhomogeneous branching Brownian motion, extending previous results by Fang and Zeitouni.

Findings

Established the convergence in law of the centered maximum M_T - m_T.

Derived an explicit formula for m_T involving T, T^{1/3}, and logarithmic terms.

Connected the asymptotics to the zeros of the Airy function.

Abstract

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \sigma(t/T), where \sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_\sigma T is negative of order T^{-1/3}, where v_\sigma is the integral of the function \sigma over the interval [0,1]. In this paper, we refine we refine this result and show the existence of a function m_T, such that M_T-m_T converges in law, as T\to\infty. Furthermore, m_T=v_\sigma T - w_\sigma T^{1/3} - \sigma(1)\log T + O(1) with w_\sigma = 2^{-1/3}\alpha_1 \int_0^1 \sigma(s)^{1/3} |\sigma'(s)|^{2/3}\,\dd s. Here, -\alpha_1=-2.33811... is the largest zero of the Airy function. The proof uses a mixture of probabilistic and analytic arguments.