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

Purdue University, West Lafayette IN

Investigators

Abstract

The PIs' research program in stochastic analysis, as part of NSF's efforts in Analysis, Modeling, and Computation of Stochastic Systems, ranges widely in probability theory and its applications to physical systems. It focuses on models in continuous space and time, with turbulent or otherwise chaotic behavior, and makes heavy use of infinite-dimensional random objects, especially stochastic partial differential equations (SPDEs) which feature white-noise behavior in time and various irregular spatial behaviors, as well as non-white-noise-based objects which fail to have the martingale or the Markov properties (e.g. fractional Brownian noise). Specific topics to be covered, with corresponding physical applications, are divided in three categories: (i) problems based on SPDEs and their probabilistic representations, including Feynman-Kac approaches, ranging from very basic questions of existence and uniqueness for highly irregular coefficients, to quantitative questions on the asymptotic behavior of linear multiplicative stochastic heat equations including the Anderson model and directed polymers measures in Gaussian environments, to questions of diffusive behavior around random Gibbsian impurities/obstacles; (ii) specific physically motivated SPDEs: magneto-hydrodynamics (MHD) in a turbulent environment, based on a Feynman-Kac formulation and its connection to products of random matrices; a framework for self-organized criticality unifying microscopic and macroscopic time scales; (iii) extensions of the Russo-Vallois theory of stochastic integration to general Gaussian and even highly non-Gaussian processes, with SPDE applications to a genealogical framework for connecting fractional Brownian motion to Kolmogorov operators. The PIs' purpose for studying these topics is to come to a better understanding of complex random ("stochastic") phenomena that change simultaneously in space and time. While many typically think of chaotic phenomena as being devoid of the possibility of predictable behavior, the PIs choice of complex models is designed to illustrate how specific inputs, no matter how random, invariably cause outputs which, while they may look very random on a short time scale, do show extremely predictable behavior in other scales of space and time, with important physical consequences. For instance, the MHD model should be capable of exhibiting the so-called "fast dynamo" effect, by which a magnetic fluid with low viscosity (the earth's oceans, or its atmosphere, or the sun), when subjected to a uniformly random energy input, will exhibit a magnetic intensity which grows at a specific exponentially rate; this effect could have applications to non-mechanical locomotion. Also of note is the model for self-organized criticality, which can help understand two-time-phased systems, such as avalanches: rather than being considered as events which occurs instantaneously when a threshold is reached, the model will take advantage of a heat-transfer setting reacting to a random environment in a short time scale. Many of the project's other models are also based on the idea that a random environment can have predictable effects, such as non-diffusive behavior for polymers or particles around random impurities or force fields. As mathematicians, the PIs are motivated by the beauty of the continuous-time continuous-space probabilistic tools needed to study these physical models, and remain true to their commitment to bridging the gap between theory and applications. Graduate students working with the PIs will take part in this project's fundamental aspects, and in investigating quantitative issues via calculations or numerical computer work. The PIs will encourage students from underrepresented groups to join their research program.