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On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes
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  • Published: 01 September 2010

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

  • M. I. Freidlin1 &
  • A. D. Wentzell2 

Probability Theory and Related Fields volume 152, pages 101–140 (2012)Cite this article

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  • 7 Citations

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    M. I. Freidlin

  2. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    A. D. Wentzell

Authors
  1. M. I. Freidlin
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  2. A. D. Wentzell
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Correspondence to M. I. Freidlin.

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Freidlin, M.I., Wentzell, A.D. On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes. Probab. Theory Relat. Fields 152, 101–140 (2012). https://doi.org/10.1007/s00440-010-0317-4

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  • Received: 20 December 2009

  • Revised: 26 June 2010

  • Published: 01 September 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0317-4

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Keywords

  • Small diffusion asymptotics
  • PDE’s with a small parameter
  • Averaging

Mathematics Subject Classification (2010)

  • 58J37
  • 60H30
  • 34C29
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