The Annals of Probability

Low-dimensional lonely branching random walks die out

Matthias Birkner and Rongfeng Sun

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Abstract

The lonely branching random walks on $\mathbb{Z}^{d}$ is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. We show that if the symmetrized walk is recurrent, lonely branching random walks die out locally. Furthermore, the same result holds if additional branching is allowed when the walk is not alone.

Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 774-803.

Dates
Received: August 2017
Revised: March 2018
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171637

Digital Object Identifier
doi:10.1214/18-AOP1271

Mathematical Reviews number (MathSciNet)
MR3916934

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
Branching random walks self-catalytic branching

Citation

Birkner, Matthias; Sun, Rongfeng. Low-dimensional lonely branching random walks die out. Ann. Probab. 47 (2019), no. 2, 774--803. doi:10.1214/18-AOP1271. https://projecteuclid.org/euclid.aop/1551171637


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References

  • [1] Birkner, M. (2003). Particle systems with locally dependent branching: Long-time behaviour, genealogy and critical parameters. Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main. Available at http://publikationen.ub.uni-frankfurt.de/volltexte/2003/314/.
  • [2] Birkner, M. and Sun, R. (2017). One-dimensional random walks with self-blocking immigration. Ann. Appl. Probab. 27 109–139.
  • [3] Bramson, M., Cox, J. T. and Greven, A. (1993). Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab. 21 1946–1957.
  • [4] Cox, J. T., Fleischmann, K. and Greven, A. (1996). Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Related Fields 105 513–528.
  • [5] Dawson, D. A. (1977). The critical measure diffusion process. Z. Wahrsch. Verw. Gebiete 40 125–145.
  • [6] Engländer, J. and Kyprianou, A. E. (2004). Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 78–99.
  • [7] Felsenstein, J. (1975). A pain in the torus: Some difficulties with models of isolation by distance. Amer. Nat. 109 359–368.
  • [8] Gorostiza, L. G., Roelly, S. and Wakolbinger, A. (1992). Persistence of critical multitype particle and measure branching processes. Probab. Theory Related Fields 92 313–335.
  • [9] Gorostiza, L. G., Roelly-Coppoletta, S. and Wakolbinger, A. (1990). Sur la persistance du processus de Dawson–Watanabe stable. L’interversion de la limite en temps et de la renormalisation. In Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Math. 1426 275–281. Springer, Berlin.
  • [10] Gorostiza, L. G. and Wakolbinger, A. (1991). Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 266–288.
  • [11] Gorostiza, L. G. and Wakolbinger, A. (1994). Long time behavior of critical branching particle systems and applications. In Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems (Montreal, PQ, 1992). CRM Proc. Lecture Notes 5 119–137. Amer. Math. Soc., Providence, RI.
  • [12] Harris, S. C. and Roberts, M. I. (2017). The many-to-few lemma and multiple spines. Ann. Inst. Henri Poincaré Probab. Stat. 53 226–242.
  • [13] Kallenberg, O. (1977). Stability of critical cluster fields. Math. Nachr. 77 7–43.
  • [14] Kurtz, T. G. and Nappo, G. (2011). The filtered martingale problem. In The Oxford Handbook of Nonlinear Filtering 129–165. Oxford Univ. Press, Oxford.
  • [15] Kurtz, T. G. and Rodrigues, E. R. (2011). Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 939–984.
  • [16] Liggett, T. M. and Port, S. C. (1988). Systems of independent Markov chains. Stochastic Process. Appl. 28 1–22.
  • [17] Liggett, T. M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 443–468.
  • [18] López-Mimbela, J. A. and Wakolbinger, A. (1997). Which critically branching populations persist? In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 203–216. Springer, New York.
  • [19] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [20] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer, New York.