Verzweigende Irrfahren in Wahrscheinlichkeitstheorie, Analysis, Algebra und Geometrie
Final Report Abstract
The purpose of this project was to work in the interdisciplinary field at the confluent of analysis, probability theory, and structure theory. One of the main objectives was to include the theory of branching random walks in the currently very active area regarding random walks on infinite graphs and groups. A branching random walk is a system of particles that move on a graph, respectively on a Cayley graph of a group. In the simplest case this model is defined as follows: the process starts with one random walker in some starting position. In each vertex of the graph the random walker produces offspring random walker that move independently of the previous history of the process. A first natural question is whether the starting position is visited infinitely many times. If the latter happens with positive probability we say the branching random walk is weakly recurrent and if it happens with probability one then it is called strongly recurrent; otherwise it is called transient. It follows from the well known result of H. Kesten that on amenable groups every (non trivial) branching random walk is strongly recurrent, while on non-amenable groups there exists a transient regime. Furthermore, N. Gantert and S. Müller gave a general criterion for transience that involves the spectral radius of the random walk. One objective was to find an analogue criterion for strong recurrence. While we failed to find such a nice criterion, we developed methods that seem to work in any concrete case. In particular, in homogeneous cases weak and strong recurrence coincide. It turns out that branching random walks are a special type of multi-type Galton–Watson processes. In this setting one is interested in criteria for global and weak/strong local survival. Observe, hereby that weak/strong local survival corresponds to weak/strong recurrence in terms of branching random walks. As for the latter phase transition we were interested in finding a nice criterion for global survival. We managed (together with N. Gantert, S. Popov, and M. Vachkovskaia) to give a criterion for one-dimensional branching random walk in random environment in terms of Lyapunof exponents of products of random matrices. While for weak/strong local survival the spectral radius of the first moments is the crucial characteristics we believe that for global extinction/survival the upper Collatz–Wieland number is a good candidate. This number also played a crucial role in a joint work with L. Gilch that studies typical random walk questions (as transience, recurrence, ergodicity, rate of escape, asymptotic entropy) for random walk on directed covers of (infinite) graphs. A further line of research went into random walk theory. Jointly with C. Gallesco, S. Popov, and M. Vachkovskaia, we studied certain systems of random walks, called spider walks, with respect to qualitative characteristics as recurrence, transience, and positive rate of escape.
Publications
-
Recurrence for branching random walks, Electronic Communications in Probability, 13:576–605, 2008
S. Müller
