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

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] Julien Berestycki, Éric Brunet, and Bernard Derrida, Exact solution and precise asymptotics of a Fisher–KPP type front, Journal of Physics A: Mathematical and Theoretical 51 (2017), no. 3, 035204.
  • [2] Julien Berestycki, Eric Brunet, and Sarah Penington, Global existence for a free boundary problem of Fisher-KPP type, arXiv:1805.03702 (2018).
  • [3] Eric Brunet and Bernard Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E 56 (1997), 2597–2604.
  • [4] John Rozier Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984, With a foreword by Felix E. Browder.
  • [5] Gioia Carinci, Anna De Masi, Cristian Giardinà, and Errico Presutti, Hydrodynamic limit in a particle system with topological interactions, Arab. J. Math. (Springer) 3 (2014), no. 4, 381–417.
  • [6] Gioia Carinci, Anna De Masi, Cristian Giardinà, and Errico Presutti, Free boundary problems in PDEs and particle systems, Springer Briefs in Mathematical Physics, vol. 12, Springer, [Cham], 2016.
  • [7] Zhen-Qing Chen, Yan-Xia Ren, and Renming Song, $l\log{L} $ criterion for a class of multitype superdiffusions with non-local branching mechanisms, arXiv:1708.08219 (2017).
  • [8] Anna De Masi and Pablo A. Ferrari, Separation versus diffusion in a two species system, Braz. J. Probab. Stat. 29 (2015), no. 2, 387–412.
  • [9] Anna De Masi, Pablo A. Ferrari, and Errico Presutti, Symmetric simple exclusion process with free boundaries, Probab. Theory Related Fields 161 (2015), no. 1-2, 155–193.
  • [10] Anna De Masi, Pablo A. Ferrari, Errico Presutti, and Nahuel Soprano-Loto, Hydrodynamics of the $ N $-BBM process, Stochastic Dynamics out of Equilibrium, Springer Proc. Math. Stat., vol. 282, Springer, 2019, pp. 523–549. arXiv:1707.00799.
  • [11] Rick Durrett and Daniel Remenik, Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations, Ann. Probab. 39 (2011), no. 6, 2043–2078.
  • [12] Nobuyuki Ikeda, Masao Nagasawa, and Shinzo Watanabe, Branching Markov processes. II, J. Math. Kyoto Univ. 8 (1968), 365–410.
  • [13] Jimyeong Lee, First passage time densities through Hölder curves, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 2, 837–849.
  • [14] Jimyeong Lee, A free boundary problem with non local interaction, Math. Phys. Anal. Geom. 21 (2018), no. 3, Art. 24, 21.
  • [15] Pascal Maillard, Speed and fluctuations of $N$-particle branching Brownian motion with spatial selection, Probab. Theory Related Fields 166 (2016), no. 3-4, 1061–1173.