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

Probability Theory and Related Fields

Volumn 152, Issue 1-2, 2012, Pages 101-140

  Freidlin, M I ^a   Wentzell, A D ^b  

^a Bldg 142   (United States)

^b Tulane University   (United States)

Author keywords

Averaging; PDE's with a small parameter; Small diffusion asymptotics

Indexed keywords

EID: 84855712312     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-010-0317-4     Document Type: Article

References (19)

1. Bochniak M.: Linear elliptic boundary value problems in varying domains. Math. Nachr. 250, 17-24 (2003).
2. Ethier S., Kurtz T.: Markov Processes, Characterization and Convergence. Wiley, London (1986).
3. Feller W.: Generalized second-order differential operators and their lateral conditions. Ill. J. Math. 1, 459-504 (1957).
4. Freidlin M.: Functional Integration and Partial Differential Equations. Princeton Univ. Press, Princeton (1985).
5. Freidlin M.: Markov Processes and Differential Equations. Birkhäuser, Basel (1996).
6. Freidlin M., Wentzell A.: Diffusion processes on graphs and the averaging principle. Ann. Probab. 21(4), 2215-2245 (1993).
7. Freidlin, M., Wentzell, A.: Random Perturbations of Hamiltonian Systems. Mem. AMS 109 (1994).
8. 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).
9. Freidlin M., Wentzell A.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Berlin (1998).
10. Freidlin M., Wentzell A.: Long-time behavior of weakly coupled oscillators. J. Stat. Phys. 123(6), 1311-1337 (2006).
11. Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983).
12. 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).
13. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland/ Kodansha, Amsterdam (1981).
14. 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).
15. Khasminskii R.: Stochastic Stability of Differential Equations, 2nd edn. Stijhoff and Noordhoff, Alphen (1980).
16. y + Cu = 0 for small ε. Ann. Math. 51, 428-445 (1950).
17. Mandl P.: Analytical Treatment of One-dimensional Markov Processes. Springer, Berlin (1968).
18. 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).
19. Molchanov S., Vainberg B.: Scattering solutions in network of thin fibers: small diameter asymptotics, Comm. Math. Phys. 273(2), 533-559 (2007).