Qualitative asymptotic problems in ergodic theory and probability

本项目的目的是从动力系统理论、遍历理论、概率论和代数等方面研究无限状态空间马氏链的行为及其相关问题的定性方面。马尔可夫动力学(确定性动力学的推广)通过从与先前状态相关联的转移概率分布中采样每个连续状态来产生随机序列。这个问题是一般伞形区域“代数、几何和组合结构上的动力系统”的一部分；它的目的是将确定性或随机演化的性质与基础状态空间的结构特征联系起来，并利用前者来阐明后者。虽然这一领域目前非常活跃(约占过去15-20年菲尔兹奖和阿贝尔奖的三分之一)，但在加拿大，除了不列颠哥伦比亚省和多伦多大学外，几乎没有这一领域。大量的结构和现象是由大型有限网络(或数学语言中的“图”)描述的。为了理解它们，人们必须研究单个元素的行为以及整个网络的结构。通常的“归纳”方法是用越来越大的有限系统来“自下而上”地逼近大系统。另一种“从上方”投影逼近无限系统的方法是香农信息论的基本原理。在香农信息论中，研究符号(“文本”)的长而有限的字符串是通过传递到从平移不变概率分布中抽样的无限字符串来进行的。一般网络也有类似的不变性概念，这相当于它们的随机同质性：所有网络节点的邻域在统计上看起来应该是相同的。这种方法与“单模随机图”的概率研究和组合学中的“图极限理论”密切相关。该项目的一部分在于进一步研究无限网络上的不变测度。我将研究这一问题的理论方面，以及此类措施的实际例子。上述网络上的不变度量也可以解释为适当定义的随机游动的平稳分布。有限状态马尔可夫链不存在的新方面是，这些随机游动可能在无穷远处表现出与非平凡边界(无论是拓扑学的还是概率论的，如泊松边界)相关的丰富行为。研究随机均匀环境中随机游动的边界(特别是在完全均匀的情况下，关于群)以及基础图和群的相关定性性质是我项目的第二部分。我希望这些边界性质也能揭示大型有限图上马尔可夫链的长时间行为。

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.