Branching Brownian motion with self repulsion

TL;DR

This paper introduces a solvable model of branching Brownian motion with self-repulsion, analyzing particle behavior, branching times, and the maximum position through reaction-diffusion equations, revealing universal properties in the weak penalty limit.

Contribution

It develops a simplified, nearly exactly solvable model of self-repelling branching Brownian motion and characterizes its particle dynamics and maximum position behavior.

Findings

Derived a precise description of particle numbers and branching times.

Identified a universal time-inhomogeneous branching process in the weak penalty limit.

Connected the maximum position to a reaction-diffusion equation with a time-dependent reaction term.

Abstract

We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term.