Asymptotic Markov dynamics

对于通常的确定性动力学，下一个位置完全由前一个位置决定。在马尔可夫动力学的情况下，这种相关性具有概率特征，因此，不是明确定义的位置，而是具有赋予概率分布(转移概率)的可能的下一个位置的“云”。在提出的研究项目中，我们感兴趣的是马尔可夫动力学的轨迹可能表现出不同的大时间行为模式的情况，例如收敛到状态空间某一边界的不同点的情况。事实上，有一种方法可以定义 抽象边界(称为马尔可夫链的泊松边界)，它描述了无穷远处轨迹的所有随机重要行为。泊松边界和某些数值不变量，如逃逸率(它描述轨迹走向无穷远的速度)或渐近熵(它描述轨迹位置提供的信息的增长率)，是这个项目的主要研究对象。在离散状态空间上的完全齐次动力学的最简单情况下的情况是众所周知的(它对应于离散群上的所谓随机游动)。我们将专注于对同质程度较低(但在合理范围内)的可能扩展。 由数学结构提供的马尔可夫动力学的类型，这些数学结构称为‘’群胚‘’。结果表明，群胚上的马尔可夫演化为众多概率模型提供了一种统一的方法。我们计划从定性和定量两个方面研究这类系统的渐近行为，并发展这种方法在分析、代数和拓扑学中的应用。

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.