Project 01: Random walk models on graphs and groups

Research interests of W. Woess The central topic of the research of W. Woess is “Random Walks on Infinite Graphs and Groups”, which is also the title of the quite successful monograph [7]. Here, Random Walks are understood as Markov chains whose transition probabilities are adapted to an algebraic, geometric, resp. combinatorial structure of the underlying state space. The main theme is the interplay between probabilisitc, analytic and potential theoretic properties of those random processes and the structural properties of that state space. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk, such as transience/recurrence, decay and asymptotic behaviour of transition probabilities, rate of escape, convergence to a boundary at infinity and harmonic functions. Vice versa, random walks may also be seen as a nice tool for classifying, or at least describing the structure of graphs, groups and related objects. The work of W. Woess is not limited to those aspects that concentrate on the link between random walks and structure theory. One one side, there is also a body of more “pure” work on infinite graphs, group actions, and also formal languages (which entered the scene via the free group). This comprises past and current collaboration with T. Ceccherini-Silberstein. On another side, some recent and less recent work concerns locally compact groups and their actions in relation with the computation of norms of transition operators, and harmonic functions on certain spaces that arise as so–called horocyclic products: past and current collaboration with S. Brofferio, M. Salvatori, L. Saloff-Coste and A. Bendikov. Woess’ research is interdisciplinary between several Mathematical areas: Probability – Graph Theory – Geometric Group Theory – Discrete Geometry – Discrete Potential Theory – Harmonic Analysis and Spectral Theory.