Quenched Tails and Almost Sure Limit Laws

0072331 Dembo这项研究是在大偏差及其应用领域。 特别是，PI将研究以一个随机性元素的固定（=淬火）实现为条件的罕见行为是确定和理解典型现象的关键的情况。 一个统一的主题是，评估罕见事件的概率导致对它们可能发生的机制的理解，当许多不同的罕见事件是可能的时，它导致识别导致其中一个实际发生的机制。 计划开展两个具体的研究方向：（一）从数学上研究和理解失去平衡的统计物理模型的动力系统的长时间行为中的“老化”现象;（二）研究布朗运动、随机游动和稳定过程等随机过程的样本路径上的例外点。 重点放在通过相应的占领措施套收缩直径定义的点。 涉及这些成分的其他研究方向适用于通用有损编码，这是一个相关的问题，对通信理论和生物分子数据分析非常感兴趣。大偏差理论主要涉及罕见事件或指数尺度上的尾部估计。 这一理论已被证明是成功的工具，推导出几乎肯定的渐近极限时，两个层次的随机性。 特别感兴趣的问题，其中大偏差的估计和想法发挥决定性作用，在确定一个极限法律，持有概率之一。

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.