Asymptotic problems for stochastic processes and corresponding partial differential equations

Abstract

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/&Alt 238; &Alt253;) of solutions to quasi-linear parabolic equations with a small parameter &Alt238;&Alt253 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 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 the PDE. In chapter 3, we study the asymptotic behavior of diffusion processes, with a small diffusion term. This process is constrained to move within some bounded domain D with instantaneous reflection on hitting the boundary ∂D of D. Such processes have applications in asymptotic questions related to linear parabolic PDEs with the Neumann boundary condition. Similar problems were previously studied by Anderson and Orey. We expand on Anderson's and Orey's work by considering different equilbra in the interior of D, similarly to the problem studied in chapter 2. However, some equilibra also appear on the boundary ∂D. We use the results of Anderson and Orey together with the work of Freidlin and Wentzell to investigate the invariant measure of the process and describe the transitions of the process between the attractors. The knowledge of the invariant measure of the process and the transition rates (in the logarithmic scale) allow us to study the long term behavior of the solution to the corresponding linear parabolic PDE as the diffusion parameter goes to zero.