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arXiv · 1507.08835

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

Bastien Mallein·Piotr Miłoś
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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 $V_n -φ\log n + o_\mathbf{P}(\log n)$, where $V_n$ 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.

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Bastien Mallein, Piotr Miłoś. 2021-12-22. Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. https://doi.org/10.1016/j.spa.2018.09.008

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