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

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

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 n, comes from time-inhomo...

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be stationary and ergodic. For A ⊂ R , let Zn(A) be the number of particles of generation n located in A. We show central l...

Annealed Moment Lyapunov Exponents for a Branching Random Walk in a Homogeneous Random Branching Environment∗

We consider a continuous-time branching random walk on the lattice Z (d ≥ 1) evolving in a random branching environment. The motion of particles proceeds according to the law of a simple symmetric random walk. The branching medium formed of Markov birth-and-death processes at the lattice sites is assumed to be spatially homogeneous. We are concerned with the long-time behavior of the quenched m...

Branching Random Walk in Random Environment: Fully Quenched Case