Asymptotic problems for stochastic partial differential equations

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The objective of this research project is the investigation and the mathematical analysis of a wide class of asymptotic problems for stochastic partial differential equations, with emphasis on the study of averaging principle, large deviation principle, stochastic singular perturbation and the interplay between them. Multi-scale and asymptotic methods in the analysis of deterministic and random systems are getting more and more important and by now they are acknowledged among the most powerful tools in many fields. They play an important role in the study of systems which possess multiple scales, in the fundamental effort to give a simplified description of them. With this project the PI is trying to apply multi-scale methods to the analysis of a large class of SPDE's: these are highly complex mathematical objects and any effort which goes in the direction of their simplification is crucial for a deeper understanding of the main features of the models and for a better effectiveness in applications. Large deviations, averaging, homogenization, singular perturbations and in general multi-scaling limits have been widely studied for systems with a finite number of degrees of freedom and a wide literature on this subject is available. The treatment of the analogous problems in the infinite dimensional case (with maybe the only exception of large deviations) is a relatively new field of investigation, which has already stirred up a vivid interest in many fields. Due to the ampleness of the subject, which is related to many branches of mathematics, and the diversified range of their applications in physics, biology and engineering, these researches are only at the beginning and a lot of work has still to be done. The analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics, from stochastic analysis, to the theory of partial differential equations, to analysis in infinite dimensional spaces.