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

Bastien Mallein and Piotr Miłoś

Stochastic Processes and their Applications, 2019, vol. 129, issue 9, 3239-3260

Abstract: The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form Vn−φlogn+oP(logn), where Vn is a function of the environment that behaves as a random walk and φ>0 is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.

Date: 2019
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DOI: 10.1016/j.spa.2018.09.008

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