zbMATH — the first resource for mathematics

On reflected Dirichlet spaces. (English) Zbl 0767.60073
Summary: Reflecting diffusion processes on smooth domains in Euclidean space are well understood. M. L. Silverstein [Ill. J. Math. 18, No. 2, 310- 355 (1974; Zbl 0303.60066) and Symmetric Markov processes (1974; Zbl 0296.60038)] developed two variant procedures for constructing the reflected processes for a general class of symmetric Hunt processes from a Dirichlet space point of view. A direct approach is given in this paper and these two variant procedures are shown to yield the same result. Only the techniques of martingales and ordinary Markov processes are used.

60J60 Diffusion processes
60G44 Martingales with continuous parameter
Full Text: DOI
[1] Albeverio, S., Ma, Z.M.: Necessary and sufficient conditions for the existence of m-perfect processes associated with Dirichlet forms. In: Azéma, J., Meyer, P.A., Yor, M. (eds.), Séminaire de Probabilités XXV (Lect. Notes Math., vol. 1485, pp. 374-406) Berlin Heidelberg: Springer 1991 · Zbl 0752.60062
[2] Albeverio, S., Ma, Z.M., Röckner, M.: A Beurling-Deny type structure theory for Dirichlet forms on general state space. BiBoS-Preprint 452 (1991); In: Proc. Memory R. H-Kroda (to appear) · Zbl 0768.31008
[3] Benveniste, A., Jacod, J.: Systems de Lévy des processus de Markov. Invent. Math.21, 183-198 (1973) · Zbl 0265.60074 · doi:10.1007/BF01390195
[4] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York London: Academic Press 1968 · Zbl 0169.49204
[5] Chen, Z.Q.: Stochastics analysis on extended sample space and tightness results. Probab. Theory Relat. Fields86, 517-549 (1990) · Zbl 0719.60076 · doi:10.1007/BF01198173
[6] Chen, Z.Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields (to appear) · Zbl 0767.60074
[7] Dellachere, C., Meyer, P.A.: Probabilities and Potential B (translated and prepared by J.P. Wilson). Amsterdam: North-Holland 1982
[8] Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math.4, 183-215 (1967) · Zbl 0317.60033
[9] Fukushima, M.: Regular representations of Dirichlet spaces. Trans. Am. Math. Soc.155, 455-473 (1971) · Zbl 0248.31007 · doi:10.1090/S0002-9947-1971-0281256-1
[10] Fukushima, M.: Dirichlet forms and Markov processes. Amsterdam: North-Holland 1980 · Zbl 0422.31007
[11] Fukushima, M., Takeda, M.: A transformation of a symmetric Markov process and the Donsker-Varadhan theory. Osaka J. Math.21, 311-326 (1984) · Zbl 0542.60077
[12] Fukushima, M.: A note on irreducibility and ergodicity of symmetric processes. In: Albeverio, S., Combe, Ph., Sirugue-Collin, M. (eds.) Stochastic processes in quantum theory and stochastical physics. Proceedings, France 1981 (Lect. Notes Math., vol. 173, pp. 200-207) Berlin Heidelberg New York: Springer 1982
[13] Gong, G.L., Qian, M.P., Silverstein, M.L.: Normal derivative for bounded domains with general boundary. Trans. Am. Math. Soc.308, 785-809 (1988) · Zbl 0661.60101 · doi:10.1090/S0002-9947-1988-0951628-0
[14] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion Processes, 2nd edn. Amsterdam: North-Holland 1988 · Zbl 0495.60005
[15] Kim, J.H.: Stochastic calculus related to non-symmetric Dirichlet forms. Osaka J. Math.24, 331-371 (1987) · Zbl 0625.60087
[16] Mizohata, S.: The theory of partial differential equations. London: Cambridge University Press 1973 · Zbl 0263.35001
[17] Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math.37, 511-537 (1984) · Zbl 0598.60060 · doi:10.1002/cpa.3160370408
[18] Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equation with discontinuous coefficients. Ann. Sci. Norm Super. Pisa, III. Ser.17, 43-77 (1963) · Zbl 0116.30302
[19] Meyer, P.A.: Intégrales stochastiques. In: Séminaire de Probabilités 1. (Lect. Notes Math., vol. 39, pp. 72-162) Berlin Heidelberg New York: Springer 1967
[20] Oshima, Y.: Lecture on Dirichlet spaces. Universität Erlangen-Nürnberg (Preprint 1988) · Zbl 0729.22011
[21] Sato, K., Ueno, T.: Multidimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ.4-3, 529-605 (1965) · Zbl 0219.60057
[22] Silverstein, M.L.: The reflected Dirichlet space. Ill. J. Math.18, (No. 2) 310-355 (1974) · Zbl 0303.60066
[23] Silverstein, M.L.: Symmetric Markov processes. (Lect. Notes Math., vol. 426) Berlin Heidelberg New York: Springer 1974 · Zbl 0296.60038
[24] Silverstein, M.L.: Boundary theory of symmetric Markov processes. (Lect. Notes Math., vol. 516) Berlin Heidelberg New York: Springer 1976 · Zbl 0331.60046
[25] Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure Appl. Math.24, 147-225 (1971) · Zbl 0227.76131 · doi:10.1002/cpa.3160240206
[26] Tanaka, H.: Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J.9, 163-177 (1979) · Zbl 0423.60055
[27] Yan, J.A.: Introduction to martingales and stochastic integrals. Shanghai Science and Technology Press: Shanghai 1981
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.