On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes

Abstract

The diffusion process in a region \({G \subset \mathbb R^2}\) governed by the operator \({\tilde L^\varepsilon = \frac{\,1}{\,2}\, u_{xx} + \frac1 {2\varepsilon}\, u_{zz}}\) inside the region and undergoing instantaneous co-normal reflection at the boundary is considered. We show that the slow component of this process converges to a diffusion process on a certain graph corresponding to the problem. This allows to find the main term of the asymptotics for the solution of the corresponding Neumann problem in G. The operator \({\tilde L^\varepsilon}\) is, up to the factor e− 1, the result of small perturbation of the operator \({\frac{\,1}{\,2}\, u_{zz}}\). Our approach works for other operators (diffusion processes) in any dimension if the process corresponding to the non-perturbed operator has a first integral, and the e-process is non-degenerate on non-singular level sets of this first integral.