On the Neumann problem for PDE's with a small parameter and the corresponding diffusion processes
成果类型：
Article
署名作者：
Freidlin, M. I.; Wentzell, A. D.
署名单位：
University System of Maryland; University of Maryland College Park; Tulane University
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-010-0317-4
发表日期：
2012
页码：
101-140
关键词：
摘要：
The diffusion process in a region G subset of R-2 governed by the operator (L) over tilde (epsilon) = 1/2u(xx) + 1/2 epsilon 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 (L) over tilde (epsilon) is, up to the factor epsilon(-1), the result of small perturbation of the operator 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.
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