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

pi在随机分析方面的研究项目，作为美国国家科学基金会在随机系统的分析、建模和计算方面的努力的一部分，广泛涉及概率论及其在物理系统中的应用。它专注于连续空间和时间中的模型，具有湍流或混沌行为，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的非白噪声对象（例如分数布朗噪声）。要涵盖的特定主题以及相应的物理应用分为三类：(i)基于spd及其概率表示的问题，包括Feynman-Kac方法，范围从非常基本的高度不规则系数的存在性和唯一性问题，到关于线性乘法随机热方程的渐近行为的定量问题，包括安德森模型和高斯环境下的定向聚合物测量，以及随机gibbsiimpurity /障碍物周围的扩散行为问题；（ii）特定的物理驱动spde：湍流环境中的磁流体动力学（MHD），基于费曼-卡茨公式及其与随机矩阵乘积的联系；统一微观和宏观时间尺度的自组织临界性框架（iii）将Russo-Vallois随机积分理论扩展到一般高斯过程甚至高度非高斯过程，并将SPDE应用于将分数阶布朗运动与Kolmogorov算子联系起来的系谱框架。pi研究这些主题的目的是为了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沌现象缺乏可预测行为的可能性，但pi选择复杂模型的目的是为了说明特定的输入，无论多么随机，总是会导致输出，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模型应该能够展示所谓的“快速发电机”效应，通过这种效应，低粘度的磁流体（地球的海洋、大气或太阳）在受到均匀随机能量输入时，将表现出以特定指数速率增长的磁性强度；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，它可以帮助理解双时相系统，如雪崩：而不是被认为是当达到阈值时立即发生的事件，该模型将利用传热设置在短时间尺度上对随机环境的反应。该项目的许多其他模型也基于这样的想法，即随机环境可以产生可预测的影响，例如聚合物或颗粒在随机杂质或力场周围的非扩散行为。作为数学家，pi们被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所激励，并始终忠于他们弥合理论与应用之间差距的承诺。与pi一起工作的研究生将参与这个项目的基础方面，并通过计算或数字计算机工作来调查定量问题。pi将鼓励来自弱势群体的学生加入他们的研究项目。

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.