Multiscale Analysis of Infinite-Dimensional Stochastic Systems

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When analyzing complex systems, it is important to be able to have a simplified description of them. Oftentimes, such a simplification is realized by considering a smaller number of factors that are considered more relevant to the understanding of these systems and by neglecting all those factors that are considered less relevant. However, some factors that may appear less important at a certain time scale, turn out to play a crucial role at some longer time scale. Thus, it is fundamental to understand correctly the interplay among multiple scales of complex systems in order to have more effective models. This project will introduce and develop new methods of asymptotic analysis for stochastic partial differential equations. These are highly complex objects and any effort that goes in the direction of their simplified and more effective analysis is important, both for their deeper understanding and for the wide range of their possible applications. This analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics, from analysis in infinite-dimensional spaces to stochastic analysis and the theory of partial differential equations. The project provides research training opportunities for graduate students. The main goal of this research project is the study of several asymptotic problems for systems that are described by stochastic partial differential equations having multiple scales. Small stochastic and deterministic perturbations of a system, which have a negligible effect on a given time scale can become crucial on a longer time scale. Limit theorems in the framework of the theory of large deviation and metastability, various realizations of the averaging principle, small mass limits as in the Smoluchowski-Kramers approximation will be the objects of the research. What characterizes and unifies the present approach to all these asymptotic problems is the effort to understand how they all interplay and interact with each other.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.