Hydrodynamics of the \(N\)-BBM process

DOI10.1007/978-3-030-15096-9_18zbMATH Open1439.60080arXiv1707.00799OpenAlexW2964272285MaRDI QIDQ2181475

Author name not available (Why is that?)

Publication date: 19 May 2020

Abstract: The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in

m a t h b b 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

h o

with a left boundary

L = s u p r i n m a t h b b R : i n t_{r}^{i} n f t y h o (x) d x = 1 > - i n f t y

and let the initial particle positions be iid continuous random variables with density

h o

. We show that the empirical measure associated to the particle positions at a fixed time

t

converges to an absolutely continuous measure with density

p s i (c d o t, t)

, as

N o i n f t y

. The limit

p s i

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.

Full work available at URL: https://arxiv.org/abs/1707.00799

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