Walks and Boundaries - a Wide Range

As it is typical for the research approaches of W. Woess, this project aims at doing mathematical research at the meeting point of several fields, strongly featuring the interplay of structure theory with probability, analysis, and combinatorics. Random walks are stochastic processes which evolve in discrete time steps on a state space with a geometric, algebraic or combinatorial structure. The spirit is conveyed by the title of a 1921 paper by G. Polya, Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Straßennetz (on an exercise of probability theory concerning the random walk in a street network). Since that, the theory has developed significantly. In the present project, the state space is not always a graph (street network), and the processes may also evolve in continuous time, resp., there may be an increasing number of moving particles. The boundaries of the title complete the spaces at infintiy. In other words, they provide a way of distinguishing points at infinity. In topic A, the random processes evolve on a boundary itself: ultrametric spaces arise as boundaries of trees. They appear e.g. in mathematical biology, while here one can trace back the inspiration to theoretical physics. Those spaces carry a family of natural random processes, generated by so-called hierarchical Laplacians. The plan is to combine random perturbations of those operators with randomising the underlying tree. B. Branching random walk studies the evolution of a population which moves according to random walk, while increasing by ongoing reproduction. We intend to study the random sequence of the empirical distributions of the population and their behaviour at infinity on trees and other infinite graphs. Topic C concerns variants of Brownian motion, where the geometry is negatively curved like in Einstein`s model of the universe, and the motion encounters lines where it is disturbed. We want to describe the long range behaviour in terms of the so-called Martin boundary, and the task is to describe the latter rigorously. D. Polyharmonic functions are related with equations coming up in the theory of elasticity or in radar imaging. Little has been done concernig discrete counterparts related with random walks. Here, the plan is to address polyharmonic functions on trees and related structures. E. Counting self-avoiding walks in graphs draws its motivation from statsitical physics and comprises hard methods and famous results. Seemingly rather different from A-D, but part of the proposer`s panorama in a natural way, we plan to study these walks from the viewpoint of formal language theory, thereby providing a link with theoretical computer science. This project and its employees are expected to interact strongly with the FWF-funded doctoral program (DK) Discrete Mathematics, of which Wolfgang Woess is the speaker.

This mathematical research project has addressed questions that deal with "paths" and "boundaries" in a variety of ways, as well as complementary topics. Here, paths (walks) are paths in graphs, or "random walks" and continuous random processes such as Brownian motion. The boundaries are sets that are added to the respective structure (graph, group) at infinity as limit points. A topic treated with high success concerns self-avoiding paths in infinite, transitive graphs, especially Cayley graphs of groups. Such paths were introduced for lattice graphs by the Nobel Prize winner in chemistry Paul Fleury as theoretical models of polymer chains, and have since played a major role at the interface of combinatorics, stochastics and statistical physics. In the project, this was linked to the theory of formal languages from theoretical computer science, and a comprehensive methodology for transitive graphs with infinite end boundary was developed. A second successful topic concerns the boundary behavior of branching random walks. These are models where in a random framework a population that increases over time moves in a space (graph, group), and the boundary behavior of the associated sequence of empirical distributions was described. A third topic combines random walks with potential theory: for random walks of nearest neighbor type on infinite trees, polyharmonic functions and their boundary behavior were studied in detail. In one of the latest project publications, this was also successfully elaborated for the continuous case of the hyperbolic plane. Other topics concern, for example, boundary entropy, the exit times of random walks from quadrants in the grid, Martin boundary and quotient limit theorems for random walks on groups, typical indices of self-similar graphs in connection with automata theory, other combinatorial - graph theoretical questions, as well as asymptotics of subordinate random walks and random processes associated with Brownian motion. The topics of this project were broad and had the particular aim of giving young researchers opportunities of development within Woess' research group. In fact, there were a total of 5 project employees and a large number of project publications that were started, carried out or completed in this project.