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.
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 maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T 1 3. We conjecture that this is the worse case correction possible. We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is @math for @math and @math when @math . In the case @math , the process is the concatenation of two BBM extremal processes, as expected. In the case @math , a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler. 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. The Random Energy Model is generalized to treat arbitrary correlations between pairs of energy levels. This Generalized Random Energy Model remains soluble. The sudden freezing which occurs in the RE model can, in the generalized version, become a progressive freezing when temperature decreases Le modele d'energie aleatoire est generalise pour prendre en compte des correlations arbitraires entre paires de niveaux d'energies. Ce modele reste un modele de verre de spin resoluble. Le gel brutal du modele a energies aleatoires peut etre remplace en gel progressif quand on baisse la temperature We prove the convergence of the extremal processes for variable speed branching Brownian motions where the "speed functions", that describe the time- inhomogeneous variance, lie strictly below their concave hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at 0 and the nal time t. The proof is based on previous results for two-speed BBM obtained in Bovier and Hartung (2014) and uses Gaussian comparison arguments to extend these to the general case. 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,