The Annals of Probability

Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations

Rick Durrett and Daniel Remenik

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Abstract

We consider a branching-selection system in ℝ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether ca or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener–Hopf equations.

Article information

Source
Ann. Probab., Volume 39, Number 6 (2011), 2043-2078.

Dates
First available in Project Euclid: 17 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1321539116

Digital Object Identifier
doi:10.1214/10-AOP601

Mathematical Reviews number (MathSciNet)
MR2932664

Zentralblatt MATH identifier
1243.60066

Subjects
Primary: 60J99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 35R35: Free boundary problems 35C07: Traveling wave solutions 60F99: None of the above, but in this section

Keywords
Branching-selection system branching random walk free boundary equation Wiener–Hopf equation traveling wave solutions

Citation

Durrett, Rick; Remenik, Daniel. Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations. Ann. Probab. 39 (2011), no. 6, 2043--2078. doi:10.1214/10-AOP601. https://projecteuclid.org/euclid.aop/1321539116


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References

  • Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
  • Bérard, J. and Gouéré, J.-B. (2010). Brunet–Derrida behavior of branching-selection particle systems on the line. Comm. Math. Phys. 298 323–342.
  • Biggins, J. D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
  • Brunet, E. and Derrida, B. (1997). Shift in the velocity of a front due to a cutoff. Phys. Rev. E (3) 56 2597–2604.
  • Brunet, É., Derrida, B., Mueller, A. H. and Munier, S. (2007). Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 041104, 20.
  • Chayes, L. and Swindle, G. (1996). Hydrodynamic limits for one-dimensional particle systems with moving boundaries. Ann. Probab. 24 559–598.
  • Durrett, R. (2004). Probability: Theory and Examples, 3rd ed. Duxbury Press, Belmont, CA.
  • Durrett, R. and Mayberry, J. (2010). Evolution in predator-prey systems. Stochastic Process. Appl. 120 1364–1392.
  • Engibaryan, N. B. (1996). Convolution equations containing singular probability distributions. Izv. Ross. Akad. Nauk Ser. Mat. 60 21–48.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880–1919.
  • Gravner, J. and Quastel, J. (2000). Internal DLA and the Stefan problem. Ann. Probab. 28 1528–1562.
  • Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 20–65.
  • Kreĭn, M. G. (1962). Integral equations on a half-line with kernel depending upon the difference of the arguments. Trans. Amer. Math. Soc. (2) 22 163–288. Original (Russian): Uspekhi Mat. Nauk. 13 (1958) 3–120.
  • Landim, C., Olla, S. and Volchan, S. B. (1998). Driven tracer particle in one-dimensional symmetric simple exclusion. Comm. Math. Phys. 192 287–307.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
  • Meirmanov, A. M. (1992). The Stefan Problem. de Gruyter Expositions in Mathematics 3. de Gruyter, Berlin.
  • Paley, R. E. A. C. and Wiener, N. (1987). Fourier Transforms in the Complex Domain. American Mathematical Society Colloquium Publications 19. Amer. Math. Soc., Providence, RI.
  • Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 43–65.
  • Spitzer, F. (1957). The Wiener–Hopf equation whose kernel is a probability density. Duke Math. J. 24 327–343.
  • Wiener, N. and Hopf, E. (1931). Üeber Eine Klasse Singulärer Integralgleichungen. Sitzungber. Akad. Wiss, Berlin 696706.