Fast flow asymptotics for stochastic incompressible viscous fluids in \mathbb {R}^2$$R2 and SPDEs on graphs (original) (raw)

Title of dissertation: ASYMPTOTIC PROBLEMS FOR STOCHASTIC PROCESSES AND CORRESPONDING PARTIAL DIFFERENTIAL EQUATIONS Lucas Tcheuko, Doctor of Philosophy, 2014 Dissertation directed by: Professor Mark Freidlin & Professor Leonid Koralov Department of Mathematics We consider asymptotic problems for diffusion processes that rely on large deviations. In Chapter 2, we study the long time behavior (at times of order exp(λ/ε)) of solutions to quasi-linear parabolic equations with a small parameter ε at the diffusion term. The solution to a partial differential equation (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 ...