Monte Carlo Methods, Metastability and Stochastic Processes with Multiple Scales

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The proposed research is primarily concerned with a rigorous investigation of Monte Carlo methods, moderate and large deviations for metastable stochastic dynamical systems that may also have multiscale features. Rare events play a key role and large deviations theory deals with their estimation. The PI will consider two broad classes of problems. The first class of problems is related to a rigorous investigation and design of provably efficient Monte Carlo methods for metastable systems. The second class of problems is about large deviations and a rigorous analysis of associated Monte Carlo methods for stochastic processes with multiple scales. The problems outlined in this research proposal are partially motivated by fundamental mathematical questions and partially by generic questions in other branches of science, such as chemical physics, biology and engineering, where metastability is important.Many problems arising from chemistry and physics involve rare but significant exit events from the basin of attraction of stable states and transition events between stable states. The transitions happen on a time scale much longer than the intrinsic time scale of the dynamical system. These rare events are important, for example in conformational changes of biomolecules, chemical reactions and nucleation events during phase transitions. Even though, a large body of work exists in the chemistry, physics and mathematics literature addressing issues related to energy landscapes and metastability, this research seeks to provide new insights about the effect of multiple scale features and about issues involved in the design of algorithms. The PI will focus on a rigorous mathematical development of moderate and large deviations as well as efficient Monte Carlo methods via large deviations for systems that may have metastability and multiple features.