Limit theorems for a branching random walk in a random or varying environment

Abstract

We consider a branching random walk on the real line with a stationary and ergodic environment (ξn) indexed by time, in which a particle of generation n gives birth to a random number of particles of the next generation, which move on the real line; the joint distribution of the number of children and their displacements on the real line depends on the environment ξn at time n. Let Zn be the counting measure at time n, which counts the number of particles of generation n situated in a Borel set of the real line. For the case where the corresponding branching process is supercritical, we establish limit theorems such as large and moderate deviation principles, central and local limit theorems on the counting measures Zn, convergence of the free energy, law of large numbers on the leftmost and rightmost positions at time n, and the convergence to infinite divisible laws. The varying environment case is also considered.