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，这种依赖性具有概率特征，因此，不是一个明确定义的位置，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”。在拟议的研究项目中，我们感兴趣的情况下，马尔可夫动力学的轨迹可能会表现出不同的模式的大时间行为，例如收敛到不同的点的状态空间的某个边界。事实上，有一种方法可以定义一个 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显著行为。泊松边界沿着与某些数值不变量，如逃逸率（它描述的速度与轨迹走向无穷大）或渐近熵（它描述的增长率的信息提供的轨迹位置），是本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（它对应于所谓的离散群上的随机游动）。我们将专注于可能的扩展到更少的同质性（但在合理的范围内） 马尔可夫动力学的类型，由称为“群胚”的数学结构提供。事实证明，群胚上的马尔可夫演化为许多概率模型提供了一种统一的方法。我们计划在定性和定量方面研究这种系统的渐近行为，并开发这种方法的应用分析，代数和拓扑。

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.