CAREER: Ergodicity and Random Media

该项目的中心目标将是发展随机媒介系统产生的随机传递算子的遍历理论。遍历理论的主要问题是随机或确定性动力系统不变测度的存在性、唯一性和吸引性。在淬火随机介质设置中，不变性自然地被斜不变性所取代，吸引力自然有两个对应的对象，即向前吸引和向后吸引。主要问题是由随机线性保锥算子积生成的环的全局吸引斜不变正解的存在性问题。因此，该项目的关键点之一是开发非紧化环境下正线性环的Perron—Frobenius理论的一般模拟。这种方法将允许对广泛的无限维随机系统进行渐近分析，并将经典的马尔可夫过程遍历理论作为一个具体的例子。它在随机环境具有一定的局部化特性的情况下最有前途。它还针对经典的小化条件失效的无限维情况。当然，在这个项目下，将研究几个与主题密切相关的问题：具有随机强迫和边界条件的偏微分方程的不变测度问题；随机定向聚合物及相关抛物模型的局部化问题随机势下动作最小路径的通用性类无限维系统转移算子的正则性问题及无限维马利亚文微积分和非自适应随机分析的相关技术。多年来，数学家和物理学家对随机介质问题和遍历理论进行了深入的研究。在这个项目中，这两个领域被结合在一起。遍历理论研究在复杂系统中产生的统计模式，在这些系统中很难或不可能做出准确的预测，只有概率预测从长远来看才有意义。有许多有趣和重要的实际情况，包括那些来自物理学，生物学和经济学的动态是由环境中存在的随机性决定的，这为分析增加了另一层复杂性。特别是，统计模式本身变得随机。这个项目的主要和统一目标是描述这些随机系统长期行为中的随机统计模式，并定性和定量地理解它们形成的机制。拟议的活动包括吸引学生从本科阶段开始研究这一领域。

The central goal of the project will be to develop the ergodic theory of random transfer operators generated by random media systems. The principal questions in ergodic theory are existence, uniqueness and attraction properties of invariant measures for stochastic or deterministic dynamical systems. In the quenched random media setting, invariance is naturally replaced by skew-invariance and attraction has two natural counterparts, forward and pullback attraction. The main question is that of the existence of a global attracting skew-invariant positive solution for a cocycle generated by products of random linear cone-preserving operators. Therefore, one of the key points of the project is to develop a general analogue of the Perron--Frobenius theory for positive linear cocycles in noncompact settings. This approach will allow the asymptotic analysis for a wide class of infinite-dimensional stochastic systems and include the classical ergodic theory of Markov processes as a specific case. It is most promising in the situations where the random environment possesses certain localization properties. It is also aimed at the infinite-dimensional situations where the classical minorization conditions fail. Naturally, under this project several problems tightly related to the main topic will be studied: the problems of invariant measures for PDEs with random forcing and boundary conditions; localization issues for random directed polymers and associated parabolic models; universality classes for action-minimizing paths in random potential; the questions of regularity of transfer operators for infinite-dimensional systems and related techniques of infinite-dimensional Malliavin calculus and non-adapted stochastic analysis.Random media problems and ergodic theory have been intensively explored by mathematicians and physicists for many years. In this project, these two fields are brought together. Ergodic theory studies statistical patterns arising in complex systems where it is hard or impossible to make exact predictions, and only probabilistic predictions make sense in the long run. There are many interesting and important practical situations including those from physics, biology and economy where the dynamics is determined by the randomness present in the environment which adds another layer of complication to the analysis. In particular, the statistical patterns themselves become random. The main and unifying goal of this project is to describe the random statistical patterns in the long-term behavior of these stochastic systems and understand qualitatively and quantitatively the mechanisms of their formation. The proposed activities include attracting students to this area of research beginning at the undergraduate level.