Stochastic Processes and Interacting Particle Systems

This research proposal is about stochastic processes in interacting particle systems. In a time span of three years we plan to study processes which have been introduced in physics and mathematical biology, using new approaches and tools for which the research units have considerable expertise. Duality is a powerful technique in probability theory which is often used to study stochastic processes in different contexts, such as interacting particle systems (ex. exclusion process), population genetics (ex. Kingman coalescence and Wright-Fisher diffusion), statistical mechanics (high and low temperature Ising model). Recently, the PI has shown that the construction of a dual process can be related to the symmetries of the generator of the original process. This has lead to the identification of a new class of interacting particle systems, which has been called “inclusion process”. The method which has been introduced is general and constructive, therefore, once the symmetry properties of the system have been identified, it can be applied in the explicit construction of new dual processes in various contexts. Metastability is a dynamical phenomenon common in a wide variety of natural systems, that can be modeled by a stochastic processes defined on the phase space of a given particle system. In this context, a relevant problem concerns the computation of the typical exit time from a metastable region. To answer this question, a potential theoretic approach to metastability has been developed in the last ten years, allowing significant improvements in the precision of estimates of the mean exit times in several models.