Electronic Journal of Probability

Non local branching Brownian motions with annihilation and free boundary problems

Anna De Masi, Pablo A. Ferrari, Errico Presutti, and Nahuel Soprano-Loto

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Abstract

We study a system of branching Brownian motions on $\mathbb{R} $ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. The case of strictly local creations (the new particle is put exactly at the same position of the branching particle) was studied in [10]. In [11] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [10] and non local branching as in [11] and prove convergence in the continuum limit (when the number $N$ of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions. We use in the convergence a stronger topology than in [10] and [11] and have explicit bounds on the rate of convergence.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 63, 30 pp.

Dates
Received: 25 May 2018
Accepted: 21 May 2019
First available in Project Euclid: 21 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1561082672

Digital Object Identifier
doi:10.1214/19-EJP324

Mathematical Reviews number (MathSciNet)
MR3978213

Zentralblatt MATH identifier
07089001

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
hydrodynamic limit free boundary problems branching Brownian motion Brunet-Derrida models

Rights
Creative Commons Attribution 4.0 International License.

Citation

De Masi, Anna; Ferrari, Pablo A.; Presutti, Errico; Soprano-Loto, Nahuel. Non local branching Brownian motions with annihilation and free boundary problems. Electron. J. Probab. 24 (2019), paper no. 63, 30 pp. doi:10.1214/19-EJP324. https://projecteuclid.org/euclid.ejp/1561082672


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