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 HamiltonJacobi equations, viscosity solutions, and
optimal control theory. I will review its brief history and the rediscovery,
and generalization of it by myself and my coauthors. I will use examples
ranging from small random pertubations of ODEs to the more physically
motivated ones such as macroscopic description of multiscale microscopic
interacting particle systems.
