Large Deviations for Markov Processes and Related Variational Problems

Large deviation estimates are probabilistic limit theorems which are used to describe atypical behavior of random systems. Markov processes are a rich class of probabilistic models. Their generators constitute a link between probability, classical analysis and *linear* partial differential equations. I will describe a method for deriving large deviation estimates for a sequence of Markov processes through convergence of some *nonlinear* transforms of their generators. This allows a connection between probability and certain topics in nonlinear analysis such as Hamilton-Jacobi equations, viscosity solutions, and optimal control theory. I will review its brief history and the re-discovery, and generalization of it by myself and my co-authors. I will use examples ranging from small random perturbations of ODEs to the more physically motivated ones, such as the macroscopic description of multi-scale microscopic interacting particle systems.