Hydrodynamics of the N-BBM process

A. DE MASI; FERRARI, PABLO A.; ERRICO PRESUTTI; SOPRANO-LOTO NAHUEL

Springer Proceedings in Mathematics & Statistics

Springer

Lugar: Berlin; Año: 2019 vol. 282 p. 523 - 549

2194-1009

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:∫∞rρ(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.