AMC-SS: Stochastic analysis and random medium in continuous space and time

PIS的随机分析研究项目是NSF随机系统分析、建模和计算课程的一部分，在概率理论及其在物理系统中的应用方面涉及广泛。它专注于连续空间和时间中具有湍流或其他混沌行为的模型，并大量使用无限维随机对象，特别是具有时间白噪声行为和各种不规则空间行为的随机偏微分方程(SPDEs)，以及不具有鞅或马尔可夫性质(如分数布朗噪声)的非白噪声对象。将涉及的具体主题和相应的物理应用分为三类：(I)基于SPDE及其概率表示的问题，包括Feynman-Kac方法，范围从高不规则系数的最基本的存在和唯一性问题，到关于高斯环境中的线性乘性随机热方程(包括Anderson模型和定向聚合物措施)渐近行为的定量问题，以及关于随机Gibsian物/障碍物周围的扩散行为的问题；(Ii)特定的物理激励SPDEs：基于Feynman-Kac公式及其与随机矩阵乘积的联系的湍流环境中的磁流体动力学(MHD)；统一微观和宏观时间尺度的自组织临界性框架；(Iii)将随机积分的Russo-Vallois理论推广到一般的高斯过程，甚至是高度非高斯过程，并将SPDE应用于将分数布朗运动与Kolmogorov算子联系起来的谱系框架。PI研究这些主题的目的是更好地理解在空间和时间上同时变化的复杂随机(“随机”)现象。虽然许多人通常认为混乱现象没有可预测行为的可能性，但复杂模型的PI选择旨在说明特定的输入如何总是导致输出，尽管它们在短时间尺度上看起来非常随机，但在其他空间和时间尺度上确实显示出极其可预测的行为，具有重要的物理后果。例如，MHD模型应该能够显示所谓的“快速发电机”效应，即具有低粘度的磁性流体(地球的海洋、大气或太阳)在受到均匀随机的能量输入时，将显示出以特定指数速率增长的磁场强度；这种效应可以应用于非机械运动。同样值得注意的是自组织临界性模型，它可以帮助理解两时间相系统，如雪崩：该模型将利用在短时间尺度上对随机环境做出反应的热传递设置，而不是被视为在达到阈值时瞬时发生的事件。该项目的许多其他模型也基于这样的想法，即随机环境可以具有可预测的影响，例如随机杂质或力场周围聚合物或颗粒的非扩散行为。作为数学家，PI的动机是研究这些物理模型所需的连续时间连续空间概率工具的美丽，并忠于他们弥合理论和应用之间的鸿沟的承诺。与PIS合作的研究生将参与这个项目的基本方面，并通过计算或数字计算机工作来研究量化问题。PIs将鼓励来自代表性不足群体的学生加入他们的研究计划。

The PIs' research program in stochastic analysis, as part of NSF'sefforts in Analysis, Modeling, and Computation of Stochastic Systems,ranges widely in probability theory and its applications to physicalsystems. It focuses on models in continuous space and time, with turbulentor otherwise chaotic behavior, and makes heavy use of infinite-dimensionalrandom objects, especially stochastic partial differential equations(SPDEs) which feature white-noise behavior in time and various irregularspatial behaviors, as well as non-white-noise-based objects which fail tohave the martingale or the Markov properties (e.g. fractional Browniannoise). Specific topics to be covered, with corresponding physicalapplications, are divided in three categories: (i) problems based on SPDEsand their probabilistic representations, including Feynman-Kac approaches,ranging from very basic questions of existence and uniqueness for highlyirregular coefficients, to quantitative questions on the asymptoticbehavior of linear multiplicative stochastic heat equations including theAnderson model and directed polymers measures in Gaussian environments, toquestions of diffusive behavior around random Gibbsianimpurities/obstacles; (ii) specific physically motivated SPDEs:magneto-hydrodynamics (MHD) in a turbulent environment, based on aFeynman-Kac formulation and its connection to products of random matrices;a framework for self-organized criticality unifying microscopic andmacroscopic time scales; (iii) extensions of the Russo-Vallois theory ofstochastic integration to general Gaussian and even highly non-Gaussianprocesses, with SPDE applications to a genealogical framework forconnecting fractional Brownian motion to Kolmogorov operators. The PIs' purpose for studying these topics is to come to a betterunderstanding of complex random ("stochastic") phenomena that changesimultaneously in space and time. While many typically think of chaoticphenomena as being devoid of the possibility of predictable behavior, thePIs choice of complex models is designed to illustrate how specificinputs, no matter how random, invariably cause outputs which, while theymay look very random on a short time scale, do show extremely predictablebehavior in other scales of space and time, with important physicalconsequences. For instance, the MHD model should be capable of exhibitingthe so-called "fast dynamo" effect, by which a magnetic fluid with lowviscosity (the earth's oceans, or its atmosphere, or the sun), whensubjected to a uniformly random energy input, will exhibit a magneticintensity which grows at a specific exponentially rate; this effect couldhave applications to non-mechanical locomotion. Also of note is the modelfor self-organized criticality, which can help understand two-time-phasedsystems, such as avalanches: rather than being considered as events whichoccurs instantaneously when a threshold is reached, the model will takeadvantage of a heat-transfer setting reacting to a random environment in ashort time scale. Many of the project's other models are also based on theidea that a random environment can have predictable effects, such asnon-diffusive behavior for polymers or particles around random impuritiesor force fields. As mathematicians, the PIs are motivated by the beauty ofthe continuous-time continuous-space probabilistic tools needed to studythese physical models, and remain true to their commitment to bridging thegap between theory and applications. Graduate students working with thePIs will take part in this project's fundamental aspects, and ininvestigating quantitative issues via calculations or numerical computerwork. The PIs will encourage students from underrepresented groups to jointheir research program.