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

TL;DR

This paper investigates the extremal process of branching Brownian motion, revealing asymptotic density behavior and stable fluctuations, and connects these findings to the Fisher-KPP equation's solutions and Bramson shift.

Contribution

It provides new limit theorems for the extremal process of branching Brownian motion and links these to the asymptotics of the Fisher-KPP equation with small initial data.

Findings

Rescaled particle density converges to an exponential distribution.

Fluctuations of the density converge to a 1-stable distribution.

Analytic techniques connect branching Brownian motion with Fisher-KPP solutions.

Abstract

We consider the limiting extremal process ${\mathcal X}$ 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 ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. 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 \cite{BD1,BD2} and further developed in the present paper. The proofs of the limit theorems for ${\mathcal X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" $0<\varepsilon\ll 1$, up to terms of the order $[{(\log \varepsilon^{-1})]^{-1-\gamma}}$, with some $\gamma>0$.