Abstract
Reflected Dirichlet space for quasi-regular Dirichlet forms is presented in this paper. We show the closedness of the (active) reflected Dirichlet forms without using the first definition of reflected Dirichlet space by Silverstein and the characterization by Chen. As an application of the closedness, the closability of distorted forms are discussed. We also show the maximality of (active) reflected Dirichlet space in the class of Silverstein's extensions and consider the uniqueness problem. Only the techniques of the transfer method and the change of underlying measures are used.
Similar content being viewed by others
References
Chen, Z. Q.: 'On reflected Dirichlet spaces', Probab. Theory Related Fields 94(2) (1992), 135-162.
Chen, Z. Q.: 'First passage distributions and reflected Dirichlet spaces', 1998, unpublished.
Chen, Z. Q., Ma, Z.-M. and Röckner, M.: 'Quasi-homeomorphism of Dirichlet forms', Nagoya Math. J. 136 (1994), 1-15.
Dong, Z., Ma, Z. M. and Sun, W.: 'A note on Beurling-Deny formulae in infinite dimensional spaces', Acta Math. Appl. Sinica 13(4) (1997), 353-361.
Eberle, A.: Uniqueness and Non-uniqueness of Singular Diffusion Operators, Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik der Universität Bielefeld, Lecture Notes in Math. 1718, Springer, 1997.
Fitzsimmons, P. J.: 'Absolute continuity of symmetric diffusions', Ann. Probab. 25(1) (1997), 230-258.
Fitzsimmons, P. J.: The Dirichlet form of a gradient-type drift transformation of a symmetric diffusion, Preprint, 1998.
Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994.
Jacob, N.: Pseudo-Differential Operators and Markov Processes, Akademie Verlag, 1996.
Kawabata, T. and Takeda, M.: 'On uniqueness problem for local Dirichlet forms', Osaka J. Math. 33(4) (1996), 881-893.
Kuwae, K. and Nakao, S.: 'Time changes in Dirichlet space theory', Osaka J. Math. 28(4) (1991), 847-865.
Kuwae, K.: 'Permanent sets of measures charging no exceptional sets and the Feynman-Kac formula', Forum Math. 7(5) (1995), 575-605.
Kuwae, K.: 'On pseudo metrics of Dirichlet forms on separable metric spaces', in Probability Theory and Mathematical Statistics (Tokyo, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 256-265.
Kuwae, K.: 'Recurrence and conservativeness of symmetric diffusion processes by Girsanov transformations', J. Math. Kyoto Univ. 37(4) (1997), 735-756.
Kuwae, K.: 'Functional calculus for Dirichlet forms', Osaka J. Math. 35(3) (1998), 683-715.
Ma, Z.-M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer Universitext, 1992.
Röckner, M. and Schmuland, B.: 'Quasi-regular Dirichlet forms: Examples and counterexamples', Canad. J. Math. 47(1) (1995), 165-200.
Schilling, R. L.: 'Conservativeness and extensions of Feller semigroups', Potential Anal. 9(1) (1998), 91-104.
Silverstein, M. L.: 'The reflected Dirichlet space', Illinois J. Math. 18 (1974), 310-355.
Silverstein, M. L.: Symmetric Markov Processes, Lecture Notes in Math. 426, Springer, 1974.
Takeda, M.: 'Two classes of extensions for generalized Schrödinger operators', Potential Anal. 5(1) (1996), 1-13.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kuwae, K. Reflected Dirichlet Forms and the Uniqueness of Silverstein's Extension. Potential Analysis 16, 221–247 (2002). https://doi.org/10.1023/A:1014081124510
Issue Date:
DOI: https://doi.org/10.1023/A:1014081124510