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 $V_n-\phi \log n+o_{\mathbf{P}}(\log n)$, where $V_n$ is a function of the environment that behaves as a random walk and $\phi >0$ is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.

Updated version with corrected typos, Stochastic Process. Appl. (2018)

References in corpus (3)

• Minima in branching random walks
• Branching random walk with a random environment in time
• Brownian motion and Random Walk above Quenched Random Wall

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