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Maximal displacement of a branching random walk in time-inhomogeneous environment

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  • Mallein, Bastien

Abstract

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length n of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time n. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order n1/3, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement.

Suggested Citation

  • Mallein, Bastien, 2015. "Maximal displacement of a branching random walk in time-inhomogeneous environment," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3958-4019.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:10:p:3958-4019
    DOI: 10.1016/j.spa.2015.05.011
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    References listed on IDEAS

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    1. Aïdékon, Elie & Jaffuel, Bruno, 2011. "Survival of branching random walks with absorption," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 1901-1937, September.
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    Cited by:

    1. Mallein, Bastien & Miłoś, Piotr, 2019. "Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3239-3260.
    2. Bezborodov, Viktor & Di Persio, Luca & Krueger, Tyll, 2021. "The continuous-time frog model can spread arbitrarily fast," Statistics & Probability Letters, Elsevier, vol. 172(C).
    3. Gerold Alsmeyer & Fabian Buckmann, 2018. "Fluctuation Theory for Markov Random Walks," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2266-2342, December.

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