Fisher-KPP equation with small data and the extremal process of branching Brownian motion

We consider the limiting extremal process X of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale Z, the rescaled density of particles, which are at distance n + x from a position close to the tip of X, converges in probability to a multiple of the exponential e(x) as n -> +infinity. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a 1-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in [9,10] and further developed in the present paper. The proofs of the limit theorems for X rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of size 0 < epsilon << 1, up to terms of the order [(log epsilon(-1))](-1-gamma), with some gamma > 0. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study focuses on the limiting extremal process of particles in binary branching Brownian motion. It demonstrates the convergence of rescaled particle density to a multiple of exponential, and density fluctuations to a 1-stable random variable. The research is motivated by the connection between the Bramson shift of Fisher-KPP equation solutions and specific initial conditions.