This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.
For other recent and relevant results on branching processes in time-inhomogeneous environments, the reader is referred to @cite_0 @cite_25 @cite_16 @cite_26 @cite_7 @cite_20 @cite_18 @cite_23 .
@cite_18 @cite_26 @cite_7 @cite_0 @cite_23 @cite_16 @cite_25 @cite_20
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In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations. 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 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. 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 n1 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. We consider the distribution of the maximum MT of branching Brownian motion with time-inhomogeneous variance of the form 2 (t=T ), where ( ) is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher KolmogorovPetrovskii-Piskunov (FKPP) equation Ft(x;t) = 2 (1 t=T )Fxx(x;t)=2 +g(F (x;t)), for appropriate nonlinearities g( ). Fang and Zeitouni (2012) showed that MT v T is negative of order T 1=3 , where v = R 1 0 (s)ds. In this paper, we show the existence of a function m 0 , such that MT m 0 converges in law, as T ! 1. Furthermore, m 0 = v T w T 1=3 (1) logT +O(1) with w = 2 1=3 1 R 1 0 (s) 1=3 j 0 (s)j 2=3 ds. Here,