Probability measures on topological groups and related problems in harmonic and functional analysis

本研究的主要主题是关于群上随机漫步的渐近（即大时间）行为的某些方面。虽然一般来说，随机漫步理论具有很强的应用色彩，但本研究主要涉及该理论的基础及其与其他数学领域的联系。群体上的概率测度和随机游走理论与数学的多个领域有着重要而深刻的相互作用。除了纯粹的概率方法外，调和和泛函分析、遍历理论和拓扑群理论等数学工具也必须参与其中才能取得进展。这种相互作用经常导致重要的问题和解决方案，这些问题和解决方案不仅在概率论内部而且在概率论之外都具有相当大的兴趣。

Probability Measures on Topological Groups and Related Problems in Harmonic and Functional Analysis The main topic of this research concerns certain aspects of the asymptotic (i.e., large time) behaviour of random walks on groups. Although the theory of random walks has, generally speaking, a strongly applied flavour, this research has to do mainly with the foundations of the theory and its connections to other fields of mathematics. The theory of probability measures and random walks on groups interacts in a nontrivial and deep way with several fields of mathematics. In addition to purely probabilistic methods, mathematical tools of harmonic and functional analysis, ergodic theory, and the theory of topological groups have to be involved to make progress. This interaction very often leads to nontrivial problems and solutions which can be of considerable interest not only within but also outside the theory of probability.