A free boundary problem arising from branching Brownian motion with selection

Berestycki, Julien; Brunet, Éric; Nolen, James; Penington, Sarah

doi:10.1090/tran/8370

A free boundary problem arising from branching Brownian motion with selection
HTML articles powered by AMS MathViewer

by Julien Berestycki, Éric Brunet, James Nolen and Sarah Penington PDF

Trans. Amer. Math. Soc. 374 (2021), 6269-6329 Request permission

Abstract:

We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in the companion paper (see Julien Berestycki, Éric Brunet, James Nolen, and Sarah Penington [Brownian bees in the infinite swarm limit, 2020]). In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.

References

Julien Berestycki, Éric Brunet, James Nolen, and Sarah Penington, Brownian bees in the infinite swarm limit, arXiv preprint, arXiv:2006.06486, 2020.
Julien Berestycki, Éric Brunet, and Sarah Penington, Global existence for a free boundary problem of Fisher-KPP type, Nonlinearity 32 (2019), no. 10, 3912–3939. MR 4012576, DOI 10.1088/1361-6544/ab25af
E. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Noisy traveling waves: effect of selection on genealogies, Europhys. Lett. 76 (2006), no. 1, 1–7. MR 2299937, DOI 10.1209/epl/i2006-10224-4
É. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization, Phys. Rev. E (3) 76 (2007), no. 4, 041104, 20. MR 2365627, DOI 10.1103/PhysRevE.76.041104
Erin Beckman, Asymptotic behavior of certain branching processes, PhD thesis, Duke University, 2019.
Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749, DOI 10.1002/9780470316962
A. Bensoussan and J.-L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Méthodes Mathématiques de l’Informatique, No. 6, Dunod, Paris, 1978 (French). MR 0513618
Nathanaël Berestycki and Lee Zhuo Zhao, The shape of multidimensional Brunet-Derrida particle systems, Ann. Appl. Probab. 28 (2018), no. 2, 651–687. MR 3784486, DOI 10.1214/14-AAP1062
M. Cabezas, A. Dembo, A. Sarantsev, and V. Sidoravicius, Brownian particles with rank-dependent drifts: out-of-equilibrium behavior, Comm. Pure Appl. Math. 72 (2019), no. 7, 1424–1458. MR 3957396, DOI 10.1002/cpa.21825
Luis A. Caffarelli and Juan L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199, DOI 10.1090/S0002-9947-1995-1260199-7
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, Cham, 2019, pp. 523–549. MR 3986078, DOI 10.1007/978-3-030-15096-9_{1}8
François Delarue, Sergey Nadtochiy, and Mykhaylo Shkolnikov, Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness, arXiv:1902.05174, 2019.
Rick Durrett and Daniel Remenik, Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations, Ann. Probab. 39 (2011), no. 6, 2043–2078. MR 2932664, DOI 10.1214/10-AOP601
N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, Ann. Probab. 25 (1997), no. 2, 702–737. MR 1434123, DOI 10.1214/aop/1024404416
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Mark Freidlin, Functional integration and partial differential equations, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. MR 833742, DOI 10.1515/9781400881598
Avner Friedman, Variational principles and free-boundary problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982. MR 679313
Richard Jordan, David Kinderlehrer, and Felix Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), no. 1, 1–17. MR 1617171, DOI 10.1137/S0036141096303359
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
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. MR 3568046, DOI 10.1007/s00440-016-0701-9