Hydrodynamic limit of N-branching Markov processes

TL;DR

This paper studies the large population limit of branching-selection particle systems with general Markovian motion, showing convergence of empirical distributions to a deterministic limit and the lowest particle to a moving boundary, extending previous Brownian motion results.

Contribution

It extends the hydrodynamic limit results to a broad class of Markov processes, including Lévy and diffusion processes, with improved convergence bounds.

Findings

Empirical distribution converges to a deterministic limit conditioned on a moving boundary.

Lowest particle position converges to the moving boundary.

Results apply to a wide class of Markov processes beyond Brownian motion.

Abstract

We consider the behaviour of branching-selection particle systems in the large population limit. The dynamics of these systems is the combination of the following three components: (a) Motion: particles move on the real line according to a continuous-time Markov process; (b) Branching: at rate 1, each particle gives birth to a new particle at its current location; (c) Selection: to keep the total number of particles constant, each branching event causes the particle currently located at the lowest position in the system to be removed instantly. Starting with N $\ge$ 1 particles whose positions at time t = 0 form an i.i.d. sample with distribution $\mu$ 0 , we investigate the behaviour of the system at a further time t > 0, in the limit N $\rightarrow$ +$\infty$. Our first main result is that, under suitable (but rather mild) regularity assumptions on the underlying Markov process, the empirical distribution of the population of particles at time t converges to a deterministic limit, characterized as the distribution of the Markov process at time t conditional upon not crossing a certain (deterministic) moving boundary up to time t. Our second result is that, under additional regularity assumptions, the lowest particle position at time t converges to the moving boundary. These results extend and refine previous works done by other authors, that dealt mainly with the case where particles move according to a Brownian motion. For instance, our results hold for a wide class of L{\'e}vy processes and diffusion processes. Moreover, we obtain improved non-asymptotic bounds on the convergence speed.