# Project C5: Random walks, branching, random media

## Principal investigators

## Participating scientists

- Johannes Blank
- Hendrik Flasche
- Michael Kochler
- Prof. Dr. Matthias Löwe
- Dr. Matthias Meiners
- Matti Schneider
- Dr. Felipe Alexis Torres Tapia

## Summary

The common theme of this project are random walks or, more generally, stochastic processes. We are interested in the interplay between properties of a random walk and the characteristics of its state space. The state space itself can be a realization of a random medium or random energy landscape. Specifically, we are addressing questions about random walks in random environments, random walks on trees, branching random walks and stochastic algorithms. Markov chain techniques are common to all the subprojects.Random walks are time-homogeneous Markov chains whose transition probabilities are adapted to a given structure of an underlying state space. This structure is geometric in nature, but may, of course, stem from an algebraic background. For instance, it can be given by an infinite, locally finite graph. This examples also includes algebraic structures as groups via their Cayley graphs. The goal, from the probabilistic viewpoint, is to understand the impact of the particular structure on the behaviour of the random walk as a stochastic process. For instance, one investigates recurrence/transience, decay of the transition probabilities, the speed of escape in the transient case, convergence to a boundary at infinity, the classification of harmonic functions. Of course, there are strong connections with other branches of mathematics such as potential theory, harmonic analysis, geometry and graph theory. On the other hand, random walks may also be seen as a tool for describing the structure of graphs, groups and related objects.

Moreover, random walks are also the basic ingredient of many models for motion in physical systems, such as the Glauber dynamics, which are the starting point for stochastic optimization algorithms (e.g. Simulated Annealing, the Metropolis chain or the Gibbs sampler).

The project is subdivided into the following subprojects.

- Random walks on trees and random environments
- Weighted branching and branching random walks
- Stochastic algorithms