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

AMC-SS：连续空间和时间中的随机分析和随机介质

• 批准号: 0606615
• 负责人: Frederi Viens
• 金额: $ 37.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0606615&HistoricalAwards=false
• 关键词: AMC; SS; Stochastic; analysis; random

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.

PI在随机分析方面的研究计划，作为NSF在随机系统分析，建模和计算方面的一部分，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，具有混沌或其他混沌行为，并大量使用无限维随机对象，特别是随机偏微分方程（SPDE），其特征是时间上的白噪声行为和各种不规则的空间行为，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）。所涉及的具体主题以及相应的物理应用分为三类：（i）基于随机微分方程及其概率表示的问题，包括Feynman-Kac方法，从高度不规则系数的存在性和唯一性的非常基本的问题，到线性乘性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中的定向聚合物测度，（ii）具体的物理驱动的SPDE：湍流环境中的磁流体力学（MHD），基于Feynman-Kac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随机积分理论推广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说明特定的输入，无论多么随机，总是会导致输出，虽然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表现出非常可预测的行为，具有重要的物理后果。例如，MHD模型应该能够解释所谓的“快速发电机”效应，即低粘度的磁性流体（地球的海洋，或大气层，或太阳），当受到均匀随机的能量输入时，将表现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，它可以帮助理解两个时间阶段的系统，如雪崩：而不是被认为是当达到阈值时立即发生的事件，该模型将利用热传递设置对短时间尺度内的随机环境做出反应。该项目的许多其他模型也基于随机环境可以产生可预测效果的想法，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，并忠于他们的承诺，弥合理论和应用之间的差距。研究生将参与这个项目的基本方面，并通过计算或数值计算机工作来调查定量问题。PI将鼓励来自代表性不足群体的学生加入他们的研究计划。