Hydrodynamics of the N-BBM Process

Abstract

The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in R and each particle at rate 1 creates a new particle at her current position; the newborn particle increments and branchings are independent of the other particles. The N-BBM starts with N particles and at each branching time, the leftmost particle is removed so that the total number of particles is N for all times. The N-BBM was proposed by Maillard and belongs to a family of processes introduced by Brunet and Derrida. We fix a density ρ with a left boundary L = sup{r ∈ R : ∞ r ρ(x)dx = 1} > −∞ and let the initial particle positions be iid continuous random variables with density ρ. We show that the empirical measure associated to the particle positions at a fixed time t converges to an absolutely continuous measure with density ψ(·, t), as N → ∞. The limit ψ is solution of a free boundary problem (FBP) when this solution exists. The existence of solutions for finite time-intervals has been recently proved by Lee.

Global existence for a free boundary problem of Fisher–KPP type

“…We construct the solution as the limit of a sequence (u n ) n≥1 , where each u n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi et al [5] show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.…”

“…existence of a solution on a time interval [0, T ] for some T > 0), under the additional assumptions that ω is absolutely continuous with respect to Lebesgue measure with probability density φ ∈ C 2 c (R), and that there exists µ 0 ∈ R such that φ(µ 0 ) = 0, φ ′ (µ 0 ) = 1 and ∞ µ 0 φ(x) dx = 1. In [5], De Masi et al study the hydrodynamic limit of the N -BBM and its relationship with the free boundary problem (FBP ′ ). The N -BBM is a variant of branching Brownian motion in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches.…”

“…We construct the solution as the limit of a sequence (u n ) n≥1 , where each u n is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi et al [5] show that this global solution can be identified with the hydrodynamic limit of the so-called N -BBM, i.e. a branching Brownian motion in which the population size is kept constant equal to N by killing the leftmost particle at each branching event.• If v (1) ≤ v (2) are two valid initial conditions and (u (i) , µ (i) ) is the solution with initial condition v (i) , then u (1) ≤ u (2) and µ (1) ≤ µ (2) .We say that (u, µ) is a classical solution to (FBP) above if (u, µ) satisfies the equation (FBP), and u(·, t) → v(·) in L 1 loc as t ց 0.…”

“…As discussed below, the condition that v is non-increasing can be relaxed to some extent.Our motivation for studying the problem (FBP) stems from its connection with the so-called N -BBM, a variant of branching Brownian motion in R in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches. More details are given in Section 2 below, but in a nutshell, De Masi et al [5] show that as N → ∞, under appropriate conditions on the initial configuration of particles, the N -BBM has a hydrodynamic limit whose cumulative distribution can be identified with the solution of (FBP), provided such a solution exists.The overall idea behind the proof is to construct u as the limit of a sequence of functions u n , where, for each n, u n satisfies an n-dependent non-linear equation, but where all the u n have the same initial condition. More precisely, let v : R → [0, 1] be a measurable function and, for n ≥ 2, let (u n (x, t), x ∈ R, t ≥ 0) be the solution to(1.1)For each n ≥ 2, this is a version of the celebrated Fisher-KPP equation about which much is known (see e.g.…”

“…Our motivation for studying the problem (FBP) stems from its connection with the so-called N -BBM, a variant of branching Brownian motion in R in which the number of active particles is kept constant (and equal to N ) by removing the leftmost particle each time a particle branches. More details are given in Section 2 below, but in a nutshell, De Masi et al [5] show that as N → ∞, under appropriate conditions on the initial configuration of particles, the N -BBM has a hydrodynamic limit whose cumulative distribution can be identified with the solution of (FBP), provided such a solution exists.…”