Introduction to Large deviations for stochastic processes

Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the award of the 2007 Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. Large deviation theory is a part of probability theory that deals with the description of events where a sum of random variables deviates from its mean by more than a 'normal' amount, i.e., beyond what is described by the central limit theorem. The mini-course will give an overview and glimpse of new techniques for large deviations for a class of stochastic processes.

The course will rely on the recent book by Feng & Kurtz on Large Deviations for Stochastic Processes, and one of our aims is to elaborate on some of the key ideas and to provide an overview. In the first part we will review basic large deviations techniques adapted for stochastic processes. Beginning with the work of Cramér and including the fundamental work on large deviations for stochastic processes by Freidlin and Wentzell and Donsker and Varadhan, much of the analysis has been based on change of measure techniques. However, recently another methodology for large deviations analogous to the Prohorov compactness approach to weak convergence of probability measures has been developed. The main theme of the course and the book by Feng & Kurtz is the development of this approach to large deviation theory as it applies to sequences of cadlag stochastic processes. This approach involves verification of exponential tightness and unique characterisation of the possible rate function. We conclude henceforth our first part with results on exponential tightness and give Puhalskii's analogue of the Prohorov compactness theorem.

In the second part, we focus on Markov processes and give large deviation results based on the convergence of the corresponding Fleming semigroups. The success of this approach depends heavily on the idea of a viscosity solution of a nonlinear equation. We outline how the comparison principle for Markov processes can be replaced by viscosity semigroup convergence. Having established the existence of a large deviation principle one needs to find appropriate representations of the rate function via stochastic control theory. We will provide a brief account on how effective dual representations of the rate functionals can be obtained. In the last part we shall demonstrate the effectiveness of these methods via examples of large deviation results for Markov processes including the Donsker-Varadhan theory for occupation measures and weakly interacting stochastic particles and deviations from hydrodynamic limits.

Jin Feng and Thomas G. Kurtz, Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs, vol. 131, AMS 2006