Analysis of stochastic partial differential equations with multiple scales

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In the description of complex systems, both deterministic and stochastic, it is usually important to be able to have a simplified model of those systems, in order to make their analysis more approachable. Usually, such a simplification is realized by looking at a smaller number of factors that are considered more relevant for the evolution of the system and by neglecting other factors that are considered less relevant. However, such an approximation, that can be effective on some given time interval, is not effective on longer time scales and the neglected factors turn out to play a fundamental role in the description of the systems' behavior. This research will analyze a large class of equations used in these models. These are highly complex equations and an understanding of them is crucial for a deeper understanding of the main features of the model and for a better effectiveness in applications. This research will develop new methods and techniques ranging over many fields of mathematics. Education and training will also be a major part of the project. The main goal of this research project is the analysis of limit theorems for stochastic partial differential equations having multiple scales. In particular, the PI will study some generalizations of the Smoluchowskii-Kramers approximation for systems with an infinite number of degrees of freedom and its long-time effects, as well as the validity of the averaging and the large deviation principle for some classes of stochastic partial differential equations (SPDEs). Specifically, the PI will try to understand what happens in the regime where the noise is weak and almost white in space. Moreover, she will study the convergence of SPDEs defined on narrow channels or describing stochastic incompressible viscous fluids in the whole space to a new class of SPDEs defined on graphs and open books. These asymptotic results will be important not only to provide a simplified description of some relevant multi-scale SPDEs that arise e.g. in the study of molecular motors and fluid dynamics, but also because at the limit they provide new interesting mathematical objects that are worthy of investigation. What characterizes and unifies this approach to all of these asymptotic problems is the effort to understand how they all interplay and interact one with the other.