Electronic Journal of Probability

Random walks in dynamic random environments and ancestry under local population regulation

Matthias Birkner, Jiří Černý, and Andrej Depperschmidt

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We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.

Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 38, 43 pp.

Received: 27 October 2015
Accepted: 23 April 2016
First available in Project Euclid: 26 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random walk dynamical random environment oriented percolation supercritical cluster central limit theorem in random environment logistic branching random walk

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Birkner, Matthias; Černý, Jiří; Depperschmidt, Andrej. Random walks in dynamic random environments and ancestry under local population regulation. Electron. J. Probab. 21 (2016), paper no. 38, 43 pp. doi:10.1214/16-EJP4666. https://projecteuclid.org/euclid.ejp/1464269713

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  • [1] Luca Avena, Renato Soares dos Santos, and Florian Völlering, Transient random walk in symmetric exclusion: limit theorems and an Einstein relation, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 2, 693–709.
  • [2] Luca Avena, Milton Jara, and Florian Voellering, Explicit LDP for a slowed RW driven by a symmetric exclusion process, preprint arXiv:1409.3013 (2014).
  • [3] Nick H. Barton, Frantz Depaulis, and Alison M. Etheridge, Neutral evolution in spatially continuous populations, Theoretical Population Biology 61 (2002), no. 1, 31–48.
  • [4] Stein Andreas Bethuelsen and Markus Heydenreich, Law of large numbers for random walks on attractive spin-flip dynamics, preprint arXiv:1411.3581v2 (2015).
  • [5] Matthias Birkner, Jiří Černý, Andrej Depperschmidt, and Nina Gantert, Directed random walk on the backbone of an oriented percolation cluster, Electron. J. Probab. 18 (2013), no. 80, 35.
  • [6] Matthias Birkner and Andrej Depperschmidt, Survival and complete convergence for a spatial branching system with local regulation, Ann. Appl. Probab. 17 (2007), no. 5-6, 1777–1807.
  • [7] J. Theodore Cox, Nevena Marić, and Rinaldo Schinazi, Contact process in a wedge, J. Stat. Phys. 139 (2010), no. 3, 506–517.
  • [8] Frank den Hollander, Harry Kesten, and Vladas Sidoravicius, Random walk in a high density dynamic random environment, Indag. Math. (N.S.) 25 (2014), no. 4, 785–799.
  • [9] Andrej Depperschmidt, Survival, complete convergence and decay of correlations for a spatial branching system with local regulation, Ph.D. thesis, TU Berlin, 2008.
  • [10] Richard Durrett, Oriented percolation in two dimensions, Ann. Probab. 12 (1984), no. 4, 999–1040.
  • [11] Richard Durrett, Multicolor particle systems with large threshold and range, J. Theoret. Probab. 5 (1992), no. 1, 127–152.
  • [12] Richard Durrett and David Griffeath, Contact processes in several dimensions, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 4, 535–552.
  • [13] Richard Durrett and Roberto H. Schonmann, Large deviations for the contact process and two-dimensional percolation, Probab. Theory Related Fields 77 (1988), no. 4, 583–603.
  • [14] Alison M. Etheridge, Survival and extinction in a locally regulated population, Ann. Appl. Probab. 14 (2004), no. 1, 188–214.
  • [15] Alison M. Etheridge, Some mathematical models from population genetics, Lecture Notes in Mathematics, vol. 2012, Springer, Heidelberg, 2011, Lectures from the 39th Probability Summer School held in Saint-Flour, 2009.
  • [16] Marta Fiocco and Willem R. van Zwet, Decaying correlations for the supercritical contact process conditioned on survival, Bernoulli 9 (2003), no. 5, 763–781.
  • [17] Olivier Garet and Régine Marchand, Large deviations for the contact process in random environment, Ann. Probab. 42 (2014), no. 4, 1438–1479.
  • [18] Geoffrey Grimmett and Philipp Hiemer, Directed percolation and random walk, In and out of equilibrium (Mambucaba, 2000), Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 273–297.
  • [19] Tomasz Komorowski, Claudio Landim, and Stefano Olla, Fluctuations in Markov processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345, Springer, Heidelberg, 2012, Time symmetry and martingale approximation.
  • [20] Thomas Kuczek, The central limit theorem for the right edge of supercritical oriented percolation, Ann. Probab. 17 (1989), no. 4, 1322–1332.
  • [21] Thomas M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999.
  • [22] Thomas M. Liggett, Roberto H. Schonmann, and Alan M. Stacey, Domination by product measures, Ann. Probab. 25 (1997), no. 1, 71–95.
  • [23] Thomas M. Liggett and Jeffrey E. Steif, Stochastic domination: the contact process, Ising models and FKG measures, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 223–243.
  • [24] Katja Miller, Random walks on weighted, oriented percolation clusters, ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016), 53–77.
  • [25] Thomas Mountford and Maria E. Vares, Random walks generated by equilibrium contact processes, Electron. J. Probab. 20 (2015), no. 3, 17.
  • [26] Firas Rassoul-Agha and Timo Seppäläinen, An almost sure invariance principle for additive functionals of Markov chains, Statist. Probab. Lett. 78 (2008), no. 7, 854–860.
  • [27] Frank Redig and Florian Völlering, Random walks in dynamic random environments: a transference principle, Ann. Probab. 41 (2013), no. 5, 3157–3180.
  • [28] Alain-Sol Sznitman, Slowdown estimates and central limit theorem for random walks in random environment, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 2, 93–143.