The objective of this research project is the development of new methods for the analysis of systems described by partial differential equations, in the presence of a noisy (random) perturbation and small parameters. We are interested in the description of the different behaviors of such systems, when the parameters are vanishing, and of the interplay between different limiting regimes. In particular, we will study new mathematical problems which are important for applications, as well as new effects in classical problems. The treatment of these problems for systems with an infinite number of degrees of freedom is a relatively new field of investigation, which has already stirred up a vivid interest in many fields, also because these are very complex objects and any effort that goes in the direction of their simplification is important for a deeper understanding of the main features of the models and for a better effectiveness in applications. Our analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics.
Our goal in this proposal is studying small deterministic and stochastic perturbations of a wide class of systems described by stochastic partial differential equations. As a matter of fact, small perturbations, which are negligible on one time scale, can become crucial on a larger time scale. The long-time influence of small perturbations has been considered in a number of our previous papers, and the present project has to be considered as a continuation of this program. Limit theorems, especially the large deviation theory, the averaging principle and the interplay between them, as well as several generalizations of the Smoluchowskii-Kramers approximation are our main tools. Systems with many/infinite degrees of freedom often have perturbations of different origin and different order. Long-time behavior of such perturbed systems should be described by a hierarchy of approximations. On the other hand, long-time behavior of pure deterministic systems with instabilities, under certain conditions, should be described by a stochastic process. Therefore, the natural generality for the problem is in considering both deterministic and stochastic perturbations of stochastic systems (not necessarily deterministic dynamical systems).
