Branching random walks in random environment: A survey
- König, Wolfgang
2010 Mathematics Subject Classification
- 60J80 60J55 60F10 60K37
- Multitype branching random walk, random potential, parabolic Anderson model, Feynman--Kac-type formula, annealed moments, large deviations
We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.