Quenched Tails and Almost Sure Limit Laws

本研究是关于大偏差及其应用领域的研究。特别是，PI将研究这样的情况，即以一个随机元素的固定(=猝灭)实现为条件的罕见行为是确定和理解典型现象的关键。一个统一的主题是，对罕见事件概率的评估有助于理解它们发生的机制，当许多不同的罕见事件可能发生时，它导致确定导致其中一种事件实际发生的机制。计划进行两条具体的研究路线：(I)对失去平衡的统计物理模型的动力系统大时间行为中的“老化”现象进行数学研究和理解；(Ii)研究随机过程(如布朗运动、随机游动和稳定过程)样本路径上的例外点。重点讨论了由缩径集对应的占位度量定义的点。涉及这些成分的拟议研究的其他方向也适用于通用有损编码，这是一个与通信理论和生物分子数据分析相关且非常感兴趣的问题。大偏差理论主要涉及指数尺度上的罕见事件或尾部估计。这一理论已被证明是一种成功的工具，当存在两个随机性水平时，可以导出几乎确定的渐近极限。特别令人感兴趣的问题是，大偏差、估计和想法在确定以概率一成立的极限定律方面起着决定性作用。

0072331Dembo This research is in the area of large deviations and its applications. In particular, the PI will study situations where the rare behavior conditioned on a fixed (= quenched) realization of one element of randomness is the key to determine and understand the typical phenomena. One unifying theme is that the evaluation of probabilities of rare events leads to an understanding of the mechanisms by which they can occur, and when many different rare events are possible, it leads to the identification of the mechanism causing one of them to actually occur. Two specific lines of research are planned: (i) Mathematical study and understanding of "aging" phenomenon in the large time behavior of dynamical systems for statistical physics models out of equilibrium; and (ii) Study of exceptional points on the sample path of stochastic processes such as Brownian motion, random walks and stable processes. Emphasis is put on points defined by means of the corresponding occupation measures of sets of shrinking diameters. Other directions of proposed research involving these ingredients are applicable among other things to universal lossy coding, a problem of relevance and much interest for communication theory and to biomolecular data analysis. The theory of large deviations is mainly concerned with rare events or tail estimates on an exponential scale. This theory has proven to be successful as a tool for deriving almost sure asymptotic limits when two levels of randomness are present. Of particular interest are problems in which large deviations estimates and ideas play a decisive role in determining a limit law that holds with probability one.