A free boundary problem arising from branching Brownian motion with selection

TL;DR

This paper investigates a free boundary problem linked to a particle system with branching Brownian motion and selection, establishing existence, uniqueness, and long-term behavior of solutions in the hydrodynamic limit.

Contribution

It introduces a novel free boundary PDE model derived from the Brownian bees particle system and proves key properties like existence, uniqueness, and asymptotic behavior.

Findings

Existence and uniqueness of the free boundary problem solution.

Characterization of the solution's behavior as time approaches infinity.

Connection of the PDE model to the underlying particle system.

Abstract

We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.