[Paper Review] Non local branching Brownians with annihilation and free boundary problems

Anna De Masi, Pablo A. Ferrari+2 more|arXiv (Cornell University)|Nov 17, 2017

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper studies a non-local branching Brownian motion with annihilation, where particles branch randomly in space and the leftmost particle is deleted to maintain a fixed population size. It establishes the hydrodynamic limit to a free boundary problem with explicit convergence rates using a stronger topology based on mass transport, extending prior work on local branching and non-local diffusion in particle systems.

ABSTRACT

We study a system of branching Brownian motions on $\mathbb R$ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new particle is put exactly at the same position of the branching particle), in [10] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [7] and non local branching as in [10] and prove convergence in the continuum limit (when the number $N$ of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions, local in time existence of classical solution has been proved recently in [13]. We use in the convergence a stronger topology than in [7] and [10] and have explicit bounds on the rate of convergence.

Motivation & Objective

• To extend previous models of branching Brownian motion with annihilation by incorporating non-local branching (where new particles are created at random positions) while retaining Brownian motion dynamics.
• To establish the hydrodynamic limit of the particle system to a continuum free boundary problem, proving convergence under a stronger topology than in prior works.
• To provide explicit, quantitative bounds on the rate of convergence to the limiting density, improving upon earlier results in the literature.
• To address the challenge that the edge of the particle system (the leftmost particle) does not move monotonically, which prevents reduction to classical Stefan-type problems.
• To develop and apply stochastic barriers (upper and lower processes) to control particle dynamics and ensure tightness in the hydrodynamic limit.

Proposed method

• Model the particle system as a system of $ N $ particles on $ \mathbb{R} $, where each particle performs independent Brownian motion and branches at rate 1, creating a new particle at position $ y $ with density $ p(x,y) $, where $ x $ is the parent's position.
• Implement an annihilation rule: immediately after each branching, the leftmost particle (minimum position) is removed to maintain a constant number of particles.
• Use a stronger topology than in previous works—based on mass transport (Wasserstein-type) metrics—to compare empirical measures of the particle system with the limiting density.
• Construct stochastic upper and lower barriers $ \underline{x}^{\delta,\pm}(t) $ to control the evolution of the particle system and ensure tightness in the hydrodynamic limit.
• Apply large deviation estimates and moment bounds (e.g., $ \mathbb{E}[n(T)^k] $) to control the number of branching events and particle positions over time.
• Prove convergence of the empirical measure to a solution of a free boundary problem defined by equations (1.1) and (1.2), with explicit error bounds via inequalities (A.3)–(A.4) and (A.6)–(A.12).

Experimental results

Research questions

• RQ1 How does the inclusion of non-local branching (non-local creation of new particles) affect the hydrodynamic limit of a branching Brownian motion with annihilation?
• RQ2 Can the hydrodynamic limit be established under a stronger topology than in prior works, and what are the implications for convergence rates?
• RQ3 What is the nature of the limiting free boundary problem, and how does it differ from classical Stefan or traveling wave problems?
• RQ4 How can stochastic barriers be used to control the position and number of particles in the system to ensure convergence?
• RQ5 What quantitative bounds can be derived for the rate of convergence of the particle system to the continuum limit?

Key findings

• The particle system converges in law to a limit density satisfying a free boundary problem defined by equations (1.1) and (1.2), under suitable assumptions on the initial density $ \rho_0 $ and kernel $ p(x,y) $.
• The convergence is established in a stronger topology than in [7] and [10], based on mass transport, which allows for explicit quantitative bounds on the rate of convergence.
• Explicit error bounds are derived: for any interval $ I $, the probability that the empirical measure deviates from the limit by more than $ N^{\alpha_1} $ is bounded by $ cN^{1-2\alpha_1} $, as in (A.3) and (A.4).
• With high probability (exceeding $ 1 - c_{b,T}N^{-1} $), all particles remain within $ [-N^b, N^b] $ for $ t \in [0,T] $, as shown in Theorem A.3 and Corollary A.4.
• The edge of the system (the leftmost particle) does not move monotonically, which prevents reduction to classical Stefan problems, and necessitates new analytical techniques.
• The use of stochastic barriers $ \underline{x}^{\delta,\pm}(t) $, constructed via coupling with a reference process, enables tight control of particle positions and supports the convergence proof.