Nonlinear Stochastic Perturbations of Dynamical Systems and Quasi-linear Parabolic PDE's with a Small Parameter

In this paper we describe the asymptotic behavior, in the exponential time scale, of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of corresponding diffusion processes. In particular, we discuss the exit problem and metastability for the processes corresponding to quasi-linear initial-boundary value problems.