A Durrett-Remenik particle system in $\mathbb{R}^d$

TL;DR

This paper analyzes a branching-selection particle system in multi-dimensional space, deriving a free boundary problem that describes the macroscopic density evolution and characterizing its limit behavior.

Contribution

It introduces a generalized model with mild assumptions and characterizes the empirical measure's limit via a unique free boundary problem solution.

Findings

The evolution is governed by an integro-differential free boundary problem.

The empirical measure converges in probability to the solution of this problem.

The free boundary corresponds to the minimal fitness value in the population.

Abstract

This paper studies a branching-selection model of motionless particles in $\mathbb{R}^d$, with nonlocal branching, introduced by Durrett and Remenik in dimension $1$. The assumptions on the fitness function, $F$, and on the inhomogeneous branching distribution, are mild. The evolution equation for the macroscopic density is given by an integro-differential free boundary problem in $\mathbb{R}^d$, in which the free boundary represents the least $F$-value in the population. The main result is the characterization of the limit in probability of the empirical measure process in terms of the unique solution to this free boundary problem.