Non local branching Brownian motions with annihilation and free boundary problems

Abstract

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.

The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection

“…This result indicates the validity of the second order corrections in (1.1). Other results on the hydrodynamic limit of the shape of the front were obtained in [11,12].…”

A Free Boundary Problem with Non Local Interaction

“…We are mainly interested here in the existence of solutions for a free boundary problem introduced in [12]. The particles system is an extension of the N-BBM model obtained by making the branching mechanism non local as in the case considered by Durrett and Remenik.…”

“…The particles system is an extension of the N-BBM model obtained by making the branching mechanism non local as in the case considered by Durrett and Remenik. In [12] the conjectured evolution equation is in fact…”

“…which is a combination of (1.1) (for the branching) and (1.3) for the Brownian diffusion. The results for the N-BBM model have been extended in [12] to this case, a limiting density ρ(x, t) exists and it is uniquely defined, moreover if there is a smooth solution of the free boundary problem (1.5)-(1.2)-(1.4) then this is the limit particles density of the model. As mentioned the proof of local existence of smooth solutions for (1.5)-(1.2)-(1.4) is the main result in this paper, the precise statement is the following.…”

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We prove local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida, [2] and Durrett and Remenik, [14]. The problem we consider describes the limit evolution of branching brownian particles on the line with death of the leftmost particle at each creation time as studied in [12]. We use extensively results in [5] and [15].

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