Ergodic Theory and Interacting Particle Systems

摘要分析和概率论的各个分支之间的相互作用有着悠久而富有成果的历史。本提案的重点是通过应用概率论的最新结果来分析动力系统，并使用遍历理论和其他分析分支来研究概率论中的问题，从而进一步加强这些联系。在遍历理论中有较大贡献的两个概率模型是第一通道渗透模型和理查森的增长模型。本文的工作旨在应用遍历理论的技术来研究Richardson生长模型中多重感染共存的问题以及第一通道渗透中单侧测地线的存在。一个提议的应用概率结果来研究动力系统是使用多米诺骨牌-可能的二次元漂移的有限型遍历性质。另一种是利用g测度遍历必要条件的最新进展来回答一个长期悬而未决的问题，即保证相应测度保持系统遍历的环面上的Anosov映射的连续模的必要条件。数学家长期以来一直使用概率模型来理解复杂的物理、生物和社会系统。这些模型的例子包括理解磁化过程的伊辛模型，通过性接触研究疾病传播的随机图，以及理解感染传播的理查森增长模型。本提案旨在通过使用动力系统数学领域的新技术来分析受物理学和生物学启发的概率模型，为这一领域做出贡献。

AbstractThere is a long and fruitful history of interaction betweenvarious branches of analysis and probability theory. Thisproposal focuses furthering these connections, both by applyingrecent results in probability to analyze dynamical systems, andusing ergodic theory and other branches of analysis, to studyquestions in probability. Two probabilistic models which haveseen large contributions from ergodic theory are first passagepercolation and Richardson's growth model. The work in thisproposal aims to apply techniques from ergodic theory to studyquestions of coexistence of multiple infections in Richardson'sgrowth models and the existence of one sided geodesics in firstpassage percolation. One proposed application of probabilityresults to study dynamical systems is the use of domino tilingsthe possible ergodic properties of two dimensional subshifts offinite type. Another is to use recent advances in necessaryconditions for ergodicity of g measures to answer a longstandingopen question about the necessary conditions modulus of continuityof an Anosov map on the torus which ensures ergodicity of thecorresponding measure preserving system.Mathematicians have long used probabilistic models to gainunderstanding of complex physical, biological and social systems.Examples of these models include the Ising model to understand theprocess of magnetization, random graphs to study the spread ofdisease through sexual contact, and Richardson's growth model tounderstand the spread of infection. This proposal aims tocontribute to this field by using new techniques in themathematical field of dynamical systems to analyze probabilisticmodels inspired by physics and biology.