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Pré-Publication, Document De Travail Année : 2023

Hydrodynamic limit of N-branching Markov processes

Résumé

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 ≥ 1 particles whose positions at time t = 0 form an i.i.d. sample with distribution µ 0 , we investigate the behaviour of the system at a further time t > 0, in the limit N → +∞. 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 Lévy processes and diffusion processes. Moreover, we obtain improved non-asymptotic bounds on the convergence speed.
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Dates et versions

hal-04296610 , version 1 (20-11-2023)

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Jean Bérard, Brieuc Frénais. Hydrodynamic limit of N-branching Markov processes. 2023. ⟨hal-04296610⟩
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