Slowdown in branching Brownian motion with inhomogeneous variance

Abstract

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form σ(t/T), where σis a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_σT is negative of order T^{-1/3}, where v_σis the integral of the function σover 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_σT - w_σT^{1/3} - σ(1)\log T + O(1) with w_σ= 2^{-1/3}α_1 \int_0^1 σ(s)^{1/3} |σ'(s)|^{2/3}\,\dd s. Here, -α_1=-2.33811... is the largest zero of the Airy function. The proof uses a mixture of probabilistic and analytic arguments.

A proof of convergence added in v2; details added and minor typos corrected in v3