Qualitative asymptotic problems in ergodic theory and probability

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The goal of this project is to investigate the qualitative aspects of the behaviour of infinite state space Markov chains and the related questions from the theory of dynamical systems, ergodic theory, probability and algebra. Markov dynamics (a generalization of the determinitsic one) produces random sequences by sampling each consecutive state from the transition probability distribution associated with the preceding state. This problematic is a part of a general umbrella area "Dynamical systems on algebraic, geometrical and combinatorial structures"; its aim is to link the properties of deterministic or stochastic evolution with the structural characteristics of the underlying state spaces, and use the former in order to elucidate the latter. Although this area is currently very active (it accounts for about one third of the Fields and Abel prizes awarded during the last 15-20 years), it is virtually absent in Canada with the exception of the universities of British Columbia and Toronto. Numerous structures and phenomena are described by large finite networks (or "graphs" in mathematical language). In order to understand them one has to study the behaviour of individual elements as well as the structure of the network as a whole. The usual "inductive" approach consists in approximating large systems "from below" by bigger and bigger finite ones. An alternative method of projective approximation "from above" by an infinite system is the foundational principle of Shannon's information theory in which long but finite strings of symbols ("texts") are studied by passing to infinite strings sampled from a shift invariant probability distribution. There is a similar notion of invariance for general networks as well, which amounts to their stochastic homogeneity: the neighbourhoods of all network nodes should statistically look the same. This approach is closely related to the probabilistic study of "unimodular random graphs" and to the "graph limit theory" in combinatorics. One part of the project consists in a further investigation of invariant measures on infinite networks. I will look both at the theoretical aspects of this problem and at the real world examples of such measures. The aforementioned invariant measures on networks can also be interpreted as stationary distributions of appropriately defined random walks. The novel aspect not present in finite state Markov chains is that these random walks may exhibit a rich behaviour at infinity related to the presence of non-trivial bordifications (either topological or probabilitistic, like the Poisson boundary). A study of the boundaries of the random walks in stochastically homogeneous environments (in particular, on groups, in the fully homogeneous case) and of the associated qualitative properties of the underlying graphs and groups is the second part of my project. I expect these boundary properties to shed light on the long time behaviour of Markov chains on large finite graphs as well.