The goal of this course is to give a self-contained introduction to the theory of large deviations, with a special emphasis on Donsker-Varadhan type asymptotics for Markov processes. We shall begin with some generalities on large deviations and present various classical results, such as the contraction principle, Varadhan’s lemma, and large deviation compactness. We next turn to Kifer’s criterion for the validity of the large deviation principle (LDP) and prove it with the help of the Dawson-Gärtner-Puhalski and Gärtner-Ellis theorems. Combining Kifer’s criterion with some results on large-time asymptotics for Feynman-Kac semigroups, we develop a general approach allowing one to establish the Donsker-Varadhan LDP for Markov processes. Finally, we apply the results obtained in the general framework to ordinary differential equations with random coefficients on a compact manifold.
Plan of the Course
- Generalities on large deviations
- Kifer’s theorem
- Donsker-Varadhan theory for Markov processes
- Application to ordinary differential equations with random coefficients
A. Shirikyan obtained his PhD degree from Moscow State University in 1995. He worked as a junior scientist at the Institute of Mechanics (Moscow) and as a research associate at the Heriot-Watt University (Edinburgh) before taking up a position of lecturer at the University of Paris-Sud in 2002. Shirikyan has been professor at CY Cergy Paris University since 2006 and adjunct professor at McGill University since 2017. He served as the head of department from 2008 to 2012 and as the director of the master programme from 2015 to 2019. Shirikyan’s main contributions in mathematics concern qualitative theory of hyperbolic PDEs, large-time behaviour of random dynamical systems, control theory for PDEs, and some aspects of non-equilibrium statistical mechanics.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai