Anomalous diffusion in strong cellular flows: averaging and homogenization

Abstract

This thesis considers the possible limit behaviors of a strong Hamiltonian cellular flow that is subjected to a Brownian stochastic perturbation.

Three possible limits are identified. When long timescales are considered, the limit behavior is described by classical homogenization theory. In the intermediate (finite) time case, it is shown that the limit behavior is anomalously diffusive. This means that the limit is given by a Brownian motion that is time changed by the local time of a process on the graph which is associated with the structure of the unperturbed flow lines (Reeb graph) that one obtains by Freidlin-Wentzell type averaging. Finally, we consider the case when the motion starts close to, or on, the cell boundary and derive a limit for the displacement on timescales of order proportional to some power of a small parameter with exponent between zero and one. (modulo a logarithmic correction to compensate for the slowdown of the flow near the saddle points of the Hamiltonian). The latter two cases are novel results obtained by the author and his collaborators.

We also consider two applications of the above results to associated partial differential equation (PDE) problems. Namely, we study a two-parameter averaging-homogenization type elliptic boundary value problem and obtain a precise description of the limit behavior of the solution as a function of the parameters using the well-known stochastic representation. Additionally, we study a similar parabolic Cauchy problem.