Quasi-linear equations with a small diffusion term and the evolution of hierarchies of cycles

We study the long time behavior (at times of order exp(λ/ε2)) of solutions to quasi-linear parabolic equations with a small parameter ε2 at the diffusion term. The solution to a PDE can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations. In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the time scale λ. We describe the evolution of the hierarchies with respect to λ in order to gain information on the limiting behavior of the solution of