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 $$\exp (\lambda /\varepsilon ^2$$ exp ( λ / ε 2 )) of solutions to quasi-linear parabolic equations with a small parameter $$\varepsilon ^2$$ ε 2 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 timescale $$\lambda $$ λ . We describe the evolution of the hierarchies with respect to $$\lambda $$ λ in order to gain information on the limiting behavior of the solution of the PDE.