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

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 ε− 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 ε-process is non-degenerate on non-singular level sets of this first integral.