Lucas Tcheuko (University of Maryland)

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

Abstract: We study the long time behavior (at times of order $\exp(\lambda/\varepsilon^2$)) of solutions to quasi-linear parabolic equations with a small parameter $\varepsilon^2$ 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 time scale $\lambda$. We describe the evolution of the hierarchies with respect to $\lambda$ in order to gain information on the limiting behavior of the solution of the PDE.