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

Bastien Mallein

doi:10.1016/j.spa.2015.05.011

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arXiv · 1307.4496

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

Bastien Mallein

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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 $n^{1/3}$, 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.

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Bastien Mallein. 2015-05-19. Maximal displacement of a branching random walk in time-inhomogeneous environment. https://doi.org/10.1016/j.spa.2015.05.011

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