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

TL;DR

This paper investigates the long-term behavior of solutions to quasi-linear parabolic PDEs with small parameters, focusing on metastability, exit problems, and the asymptotic behavior of related diffusion processes in exponential time scales.

Contribution

It provides new insights into the asymptotic analysis of quasi-linear parabolic equations with small parameters and their associated diffusion processes, including metastability and exit phenomena.

Findings

Characterization of the exponential time scale behavior of solutions.

Analysis of metastability and exit problems for the processes.

Description of the long-time behavior of diffusion processes.

Abstract

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.