Large deviation principle for the diffusion-transmutation processes and dirichlet problem for PDE systems with small parameter
- 62 Downloads
- 1 Citations
Summary
The diffusion-transmutation processes are considered as the diffusivities are of order ε,ε→0 and the transmutation intensities are of order ε−1. We prove a large deviation principle for the position joint with the type occupation times as ε→0 and study the exit problem for this process. We consider the Levinson case where a trajectory of the average drift field exits from a domain in finite time in a regular way and the large deviation case where the average drift field on the boundary points inward at the domain. The exit place and the type distribution at the exit time are determined as ε→0; this gives the limit of the Dirichlet problems for the corresponding PDE systems with a parameter ε→0.
Mathematics Subject Classification (1991)
60F 10 35J55Preview
Unable to display preview. Download preview PDF.
References
- [BL] BenArous, G., Ledoux: Schilder large deyiation theorem without topology. In: Asymptotic problems in probability theory, Proc. Tauiguchi Symp. 1990, Kyoto. Pitman Research Notes, 1992Google Scholar
- [D] Doob, J.L.: Stochastic processes. New York: Wiley, 1990Google Scholar
- [DV] Donsker, M., Varadhan, S.R.S.: Asymptotic evaluation of certain Wiener integrals for large time. In: Arthurs, A.M. (ed.) Functional integration and its applications, pp. 15–33. Oxford: Clarendon Press, 1975Google Scholar
- [EF1] Eizenberg, A., Freidlin, M.I.: On the Dirichlet problem for a class of second order PDE systems with small parameter. Stoch. and Stoch. Reports,33, 111–148 (1990)Google Scholar
- [EF2] Eizenberg, A., Freidlin, M.I.: PDE-systems with a small parameter. Lect. Appl. Math.27, 175–184 (1991)Google Scholar
- [F] Freidlin, M.I.: Functional integration and partial differential equations, Princeton, NJ: Princeton University Press, 1985Google Scholar
- [FW] Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Berlin, Heidelberg: Springer 1984Google Scholar
- [G] Gärtner, J.: On logarithmic asymptotics of the probabilities of large deviations. Ph. D. Thesis, Moscow State University 1976 (in Russian)Google Scholar
- [GS] Gikhman, I.I., Skorokhod: The theory of stochastic processes, Berlin, New York: Springer 1974-1979Google Scholar
- [PW] Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Berlin, Heidelberg, New York: Springer 1984Google Scholar
- [R] Rowe, E.: Recurrency and transiency of Markov processes governed by coupled systems of second-order elliptic operators. Commun. Partial Differential Equations17, 2093–2112 (1992)Google Scholar
- [V] Varadhan, S.R.S.: Large deviations and applications. Philadelphia: Society for Industrial and Applied Mathematics, 1984Google Scholar