Asymptotic Markov dynamics

渐近马尔可夫动力学

• 批准号: 402587-2011
• 负责人: Kaimanovich, Vadim
• 金额: $ 1.17万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573327
• 关键词: Asymptotic; Markov; dynamics

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.

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，这种依赖具有概率特征，因此，不是一个定义良好的位置，而是一个具有概率分布（转移概率）的可能的下一个位置的“云”。在提出的研究项目中，我们对马尔可夫动力学的轨迹可能表现出大时间行为的不同模式的情况感兴趣，例如收敛到状态空间的某个边界的不同点。事实上，有一种方法可以定义an