A Free Boundary Problem with Non Local Interaction

Lee, Jimyeong

doi:10.1007/s11040-018-9282-4

Cited by 2 publications

References 21 publications

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Hydrodynamics of the N-BBM Process

Masi

Ferrari

Presutti

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in R and each particle at rate 1 creates a new particle at her current position; the newborn particle increments and branchings are independent of the other particles. The N-BBM starts with N particles and at each branching time, the leftmost particle is removed so that the total number of particles is N for all times. The N-BBM was proposed by Maillard and belongs to a family of processes introduced by Brunet and Derrida. We fix a density ρ with a left boundary L = sup{r ∈ R : ∞ r ρ(x)dx = 1} > −∞ and let the initial particle positions be iid continuous random variables with density ρ. We show that the empirical measure associated to the particle positions at a fixed time t converges to an absolutely continuous measure with density ψ(·, t), as N → ∞. The limit ψ is solution of a free boundary problem (FBP) when this solution exists. The existence of solutions for finite time-intervals has been recently proved by Lee.

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Hydrodynamics of the N-BBM Process

Masi

Ferrari

Presutti

et al. 2019

Springer Proceedings in Mathematics &Amp; Statistics

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Non local branching Brownian motions with annihilation and free boundary problems

Masi¹,

Ferrari²,

Presutti³

et al. 2019

Electron. J. Probab.

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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.

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A Free Boundary Problem with Non Local Interaction

Cited by 2 publications

References 21 publications

Hydrodynamics of the N-BBM Process

Hydrodynamics of the N-BBM Process

Non local branching Brownian motions with annihilation and free boundary problems

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