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

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 $V_n -\varphi \log n + o_\P(\log n)$, where $V_n$ is a function of the environment that behaves as a random walk and $\varphi>0$ 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. Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. Stochastic Processes and their Applications, Elsevier, 2018, 125 (10), pp.3958-4019. ⟨10.1016/j.spa.2015.05.011⟩. ⟨hal-00874541v2⟩

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