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Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

Jacob Mercer

TL;DR

The paper analyzes a two-sided branching Brownian motion with selection, the (N,p)-BBM, and proves that its hydrodynamic limit is a two-sided free boundary problem on a finite interval with moving boundaries L_t and R_t, coupled to Neumann-type boundary data determined by p. It constructs and analyzes discrete-time upper and lower bounding processes X^{N,p,δ,±}, establishes their convergence to deterministic limits, and shows these converge to the same limit as δ→0, yielding the desired FBP solution. The work further proves that the asymptotic speed v_N of the particle system converges to the traveling-wave speed v_p of the limiting FBP, thereby extending one-sided Brunet-Derrida results to a two-sided setting. The methods combine probabilistic representations, coupling constructions, and comparison principles for FBPs, with auxiliary results on left/right boundary convergence and speed. Overall, the study links branching diffusion with two-sided selection to a well-posed free boundary framework and clarifies the large-N behavior and wave-speed selection mechanisms in this broader context.

Abstract

We introduce and analyse a two-sided free boundary problem arising from a generalization of the well-known $N$-particle branching Brownian motion ($N$-BBM) model, called the $(N, p)$-BBM, where selection can occur on both the leftmost and rightmost particles according to a parameter $p\in(0,1)$. The main result here establishes that, as $N\to\infty$, the empirical distribution of the $(N,p)$-BBM converges to a deterministic hydrodynamic limit described by a free boundary problem on a finite interval with two moving boundaries, and explicit Neumann and Dirichlet boundary conditions parametrized by $p$. We further prove that the asymptotic velocity $v_{N,p}$ of the $(N,p)$-BBM converges, as $N\to\infty$, to $v_p$, the unique travelling wave speed of the limiting free boundary problem. These results generalize previous one-sided models and connect to broader classes of free boundary problems found in evolutionary dynamics and flame propagation.

Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

TL;DR

The paper analyzes a two-sided branching Brownian motion with selection, the (N,p)-BBM, and proves that its hydrodynamic limit is a two-sided free boundary problem on a finite interval with moving boundaries L_t and R_t, coupled to Neumann-type boundary data determined by p. It constructs and analyzes discrete-time upper and lower bounding processes X^{N,p,δ,±}, establishes their convergence to deterministic limits, and shows these converge to the same limit as δ→0, yielding the desired FBP solution. The work further proves that the asymptotic speed v_N of the particle system converges to the traveling-wave speed v_p of the limiting FBP, thereby extending one-sided Brunet-Derrida results to a two-sided setting. The methods combine probabilistic representations, coupling constructions, and comparison principles for FBPs, with auxiliary results on left/right boundary convergence and speed. Overall, the study links branching diffusion with two-sided selection to a well-posed free boundary framework and clarifies the large-N behavior and wave-speed selection mechanisms in this broader context.

Abstract

We introduce and analyse a two-sided free boundary problem arising from a generalization of the well-known -particle branching Brownian motion (-BBM) model, called the -BBM, where selection can occur on both the leftmost and rightmost particles according to a parameter . The main result here establishes that, as , the empirical distribution of the -BBM converges to a deterministic hydrodynamic limit described by a free boundary problem on a finite interval with two moving boundaries, and explicit Neumann and Dirichlet boundary conditions parametrized by . We further prove that the asymptotic velocity of the -BBM converges, as , to , the unique travelling wave speed of the limiting free boundary problem. These results generalize previous one-sided models and connect to broader classes of free boundary problems found in evolutionary dynamics and flame propagation.
Paper Structure (13 sections, 27 theorems, 151 equations)

This paper contains 13 sections, 27 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Some properties of the $(N,p)$-BBM
  4. Coupling particle systems with two-sided selection
  5. Construction of $X$:
  6. Construction of $X^+$:
  7. Construction of $X^-$:
  8. Convergence of the Deterministic Limits
  9. Probabilistic representation of solutions of free boundary problems
  10. A Comparison Theorem for the Free Boundary Problem
  11. Convergence of the Leftmost and Rightmost Particles
  12. Convergence of Speeds
  13. Appendix

Key Result

Theorem 1

Consider the $(N,p)$-BBM process $(X^{N,p}(t))_{t\geq 0}$ such that, at time $t=0$, each particle independently has distribution $\rho$. Let $\pi^{N,p}_{\rho,t}$ be the empirical distribution of the $N$ particles of $(X^{N,p}(t))_{t\geq 0}$ at time $t$. Fix $t>0$. Then there exists a function $u(\cd Moreover, when the solution of the FBP (fbpMain) with initial condition $\rho$ exists and has local

Theorems & Definitions (56)

  • Theorem 1
  • Conjecture 2
  • Proposition 3
  • proof
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Proposition 7
  • proof
  • Definition 8
  • ...and 46 more