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.
References
Bochniak M.: Linear elliptic boundary value problems in varying domains. Math. Nachr. 250, 17–24 (2003)
Ethier S., Kurtz T.: Markov Processes, Characterization and Convergence. Wiley, London (1986)
Feller W.: Generalized second-order differential operators and their lateral conditions. Ill. J. Math. 1, 459–504 (1957)
Freidlin M.: Functional Integration and Partial Differential Equations. Princeton Univ. Press, Princeton (1985)
Freidlin M.: Markov Processes and Differential Equations. Birkhäuser, Basel (1996)
Freidlin M., Wentzell A.: Diffusion processes on graphs and the averaging principle. Ann. Probab. 21(4), 2215–2245 (1993)
Freidlin, M., Wentzell, A.: Random Perturbations of Hamiltonian Systems. Mem. AMS 109 (1994)
Freidlin M., Wentzell A.: Weak convergence of one-dimensional Markov processes. In: Freidlin, M. (eds) Markov Processes and Their Applications, The Dynkin Festschrift, pp. 95–111. Birkhäuser, Boston (1994)
Freidlin M., Wentzell A.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Berlin (1998)
Freidlin M., Wentzell A.: Long-time behavior of weakly coupled oscillators. J. Stat. Phys. 123(6), 1311–1337 (2006)
Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)
Grieser, D.: Thin tubes in mathematical physics, global analysis and spectral geometry. In: Exner, P., et al. (eds.) Analysis on Graphs and its Applications. Proceedings of Symposia in Pure Mathematics, pp. 565–594, AMS (2008)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland/ Kodansha, Amsterdam (1981)
Khasminskii R.: Principle of averaging for parabolic and elliptic differential equations and of Markov processes with small diffusion. Theory Probability and Appl. 8, 1–21 (1963)
Khasminskii R.: Stochastic Stability of Differential Equations, 2nd edn. Stijhoff and Noordhoff, Alphen (1980)
Levinson N.: The first boundary-value problem for the equation εΔu + A u x + B u y + C u = 0 for small ε. Ann. Math. 51, 428–445 (1950)
Mandl P.: Analytical Treatment of One-dimensional Markov Processes. Springer, Berlin (1968)
Molchanov, S., Vainberg, B.: Laplace equation in network of thin fibers. In: Ben Arous, G., et al. (eds.) Stoch. Analysis in Math. Physics, Proc. of satellite conference of ICM, pp. 69–93 (2006)
Molchanov S., Vainberg B.: Scattering solutions in network of thin fibers: small diameter asymptotics, Comm. Math. Phys. 273(2), 533–559 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0317-4
Keywords
- Small diffusion asymptotics
- PDE’s with a small parameter
- Averaging
Mathematics Subject Classification (2010)
- 58J37
- 60H30
- 34C29