MARK KAC SEMINAR

I plan to use this talk to provide a user's guide for a method described in the book "Large deviations for stochastic processes" by Feng and Kurtz (2006). This is a Hamilton–Jacobi approach to the theory of large deviations for metric space valued Markov processes.

In the first part, I will explain some basic observations that lead to such a formulation. Then I will describe the whole scheme and mathematical challenges associated with applications of the theory to concrete examples.

In the second part, I will focus on examples and the proof of comparison principles. Establishing such principles is a key step in applying the theory. The examples are taken from applications in statistical mechanics and continuum mechanics.

I will describe a Lie algebraic approach to stochastic duality for interacting particle systems. The approach, in the form I will discuss, was first introduced in a 2009 paper jointly with Kurchan, Redig, Vafayi (picking up previous ideas of Lloyd Sudbury (1995) and of Schutz (1996)) and it was then expanded in a series of works jointly with Carinci, Giberti, Redig (2013, 2014). In this novel approach, three basic ideas provide a general scheme for duality: the generator of a Markov process is recognized as an element of a (quantum) Lie algebra; dual processes are obtained by a change of representation of the algebra; self-dual processes are related to symmetries of the algebra.

I will review several examples of interacting particle systems where the scheme can be successfully applied. I will consider examples both from mathematical population genetics and from non-equilibrium statistical mechanics. Besides classical examples, I will also show the constructive power of the method, by considering new processes that have been recently introduced in a joint work with Carinci, Redig, Sasamoto (2015).