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Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach

Published online by Cambridge University Press:  14 November 2011

Z. Q. Chen  and
Z. Zhao
Z. Q. Chen
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A.
Z. Zhao
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.
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Abstract

The switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a Dirichlet space approach and is applied to prove the existence theorem of the Cauchy initial problem for the system. A representation theorem for the solution of the Dirichlet boundary value problem and a generalised Skorohod decomposition for the reflecting switched diffusion process are obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 673 - 701
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Bass, R. and Hsu, P.. Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 (1991), 486–508.CrossRefGoogle Scholar
2Benveniste, A. and Jacod, J.. Systèmes de Lévy des processus de Markov. Invent. Math. 21 (1973), 183–198.CrossRefGoogle Scholar
3Bliedtner, J.. Functional spaces and their exceptional sets. Seminar on Potential Theory, II, Lecture Notes in Mathematics 226, 1–13 (Berlin: Springer, 1970).Google Scholar
4Carrillo Menendez, S.. Processus de Markov associé à une forme de Dirichlet non symétrique. Z. Wahrsch. verw. Gebiete 33 (1975), 139–154.CrossRefGoogle Scholar
5Carrillo Menendez, S.. Sur la régularité des potentials des espaces de Dirichlet. C. R. Acad. Sci. Paris Sér. 1 Math. 302(8) (1986), 329–332.Google Scholar
6Chen, Z. Q.. On reflecting diffusion processes and Skorohod decompositions. Probab. Theory Related Fields 94 (1993), 281–315.CrossRefGoogle Scholar
7Dellacherie, C. and Meyer, P. A.. Probabilitiés et potential; Theories des Martingales (Paris: Hermann, 1980).Google Scholar
8Edmunds, D. E. and Evans, W. D.. Spectral Theory and Differential Operators (Oxford: Clarendon Press, 1987).Google Scholar
9Eizenberg, A. and Freidlin, M.. On the Dirichlet problem for a class of second order PDE systems with small parameter. Stochastics Stochastics Rep. 33 (1990), 111–148.CrossRefGoogle Scholar
10Eizenberg, A. and Freidlin, M.. Partial differential systems with a small parameter and diffusion process with a discrete component. Lecture Notes in Appl. Math. 27 (1991), 175–184.Google Scholar
11Freidlin, M.. Functional Integration and Partial Differential Equations (Princeton: Princeton University Press, 1985).Google Scholar
12Fukushima, M.. Dirichlet Forms and Markov Processes (Amsterdam: North-Holland, 1980).Google Scholar
13Habetler, G. and Martino, M.. Existence theorems and the spectral theory for the multi-group diffusion model. Proc. Sympos. Appl. Math. XI, Nuclear Reactor Theory, 127–139 (Providence, R.I.: American Mathematical Society, 1961).Google Scholar
14Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1966).Google Scholar
15Kifer, Y.. Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic systems of PDE (preprint).Google Scholar
16Kim, J. H.. Stochastic Calculus related to non-symmetric Dirichlet forms. Osaka J. Math. 24 (1987), 331–371.Google Scholar
17Kunita, H.. Sub-Markov semi-groups in Banach lattices. Proceedings of the International Conference on Functional Analysis and Related Topics, (Tokyo: University of Tokyo Press, 1969). 332–343.Google Scholar
18LeJan, Y.. Measures associées à une forme de Dirichlet. Applications. Bull. Soc. Math. France 106 (1978), 61–112.CrossRefGoogle Scholar
19Oshima, Y.. Lectures on Dirichlet spaces (preprint, Universität Erlangen-Nürnberg, 1988).Google Scholar
20Protter, M. and Weinberger, H.. Maximum Principles in Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1976).Google Scholar
21Silverstein, M. L.. Symmetric Markov Processes, Lecture Notes in Mathematics 426 (Berlin: Springer, 1974).CrossRefGoogle Scholar
22Silverstein, M. L.. Applications of the sector condition to the classification of sub-Markovian semigroups. Trans. Amer. Math. Soc. 244 (1978), 103–146.Google Scholar
23Simon, B.. Schrödinger semigroup. Bull. Amer. Math. Soc. 7 (1982), 447–526.CrossRefGoogle Scholar
24Skorohod, A. V.. Asymptotic methods of the theory of Stochastic differential equations, Translations of Mathematical Monographs 78 (Providence, R.I.: American Mathematical Society, 1989).Google Scholar
25Sweers, G.. Strong positivity in C for elliptic systems. Math. Z. 209 (1992), 251–271.CrossRefGoogle Scholar