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

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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.