Non local branching Brownians with annihilation and free boundary problems

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

ELECTRONIC JOURNAL OF PROBABILITY

UNIV WASHINGTON

Año: 2019 vol. 24 p. 1 - 30

1083-6489

We study a system of branching Brownian motions on R with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new particle is put exactly at the same position of the branching particle), in [10] 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 [7] and non local branching as in [10] 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, local in time existence of classical solution has been proved recently in [13]. We use in the convergence a stronger topology than in [7] and [10] and have explicit bounds on the rate of convergence.