Prof. Dr. Wolfgang König
Duration: 01.04.2013 - 31.08.2016
We study the long-time behaviour of branching random walk in random environment (BRWRE) on the d-dimensional lattice. We consider one of the basic models, which includes migration and branching/killing of the particles, given a random potential of spatially dependent branching/killing rates. Based on the observation that the expectation of the population size over the migration and the branching and killing is equal to the solution to the well-known and much-studied parabolic Anderson model (PAM), we will use our understanding of the long-time behaviour of the PAM to develop a detailed picture of the BRWRE. Furthermore, we will exploit methods that were successful in the treatment of the PAM to prove at least part of this picture. Particular attention is payed to the study of the concentration of the population in sites that determine the long-time behaviour of the PAM, which shows a kind of intermittency. One fundamental thesis that we want to make precise and rigorous is that the overwhelming contribution to the total population size of the BRWRE comes from small islands where most of the particles travel to and have a extremely high reproduction activity. We aim at a detailed analysis for the case of the random potential being Pareto-distributed, in which case the rigorous study of the PAM has achieved a particularly clear picture. This project has the following four main goals. I. For a variety of random potentials, we derive large-time asymptotics for the n-th moments of the local and total population size, based on techniques from the study of the PAM. II. We want to understand and identify the limiting distributions of the global population size by a finer analysis for Pareto-distributed potentials. III. We want to investigate, for Pareto-distributed potentials, the long-time (de)correlation properties of the evolution of the particles such as aging, in particular, slow/fast evolution phenomena and what type of aging functions will appear. IV. We want to study, for Paretodistributed potentials, almost surely with respect to the potential, the particle flow of the BRWRE in a geometric sense by finding trajectories along which most of the particles travel and branch, in particular the sites and the time intervals where, respectively when, most of the particles show an extremely high reproduction activity.