Asymptotic Markov dynamics

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For the usual deterministic dynamics the next position is completely determined by the previous one. In the case of Markov dynamics this dependence has a probabilistic character, so that instead of a well-defined position one rather has a ``cloud'' of possible next positions endowed with a probability distribution (a transition probability). In the proposed research project we are interested in the situations when the trajectories of a Markov dynamics may exhibit different patterns of large time behaviour, for instance converge to different points of a certain boundary of the state space. In fact, there is a way to define an abstract boundary (called the Poisson boundary of a Markov chain) which describes all stochastically significant behaviour of trajectories at infinity. The Poisson boundary along with certain numerical invariants, like the rate of escape (it describes the speed with which the trajectories go to infinity) or the asymptotic entropy (it describes the growth rate of the information provided by trajectory positions), are the principal objects of study in this project. The situation in the simplest case of a completely homogeneous dynamics on a discrete state space is well-known (it corresponds to the so-called random walks on discrete groups). We are going to concentrate on possible extensions to less homogeneous (within reasonable limits though) types of Markov dynamics which are provided by the mathematical structures called ``groupoids''. It turns out that Markov evolutions on groupoids provide a unified approach to numerous probabilistic models. We plan to study the asymptotic behaviour of such systems in both qualitative and quantitative aspects, and to develop applications of this approach to analysis, algebra and topology.