Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Regular representations of Dirichlet spaces


Author: Masatoshi Fukushima
Journal: Trans. Amer. Math. Soc. 155 (1971), 455-473
MSC: Primary 60.60
DOI: https://doi.org/10.1090/S0002-9947-1971-0281256-1
MathSciNet review: 0281256
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space.


References [Enhancements On Off] (What's this?)

  • [1] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208-215. MR 21 #5098. MR 0106365 (21:5098)
  • [2] J. Deny, Principe complet du maximum et contractions, Ann. Inst. Fourier (Grenoble) 15 (1965), fasc. 1, 259-272. MR 32 #5913. MR 0188475 (32:5913)
  • [3] J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 305-370. MR 17, 646. MR 0074787 (17:646a)
  • [4] J. L. Doob, Boundary properties of functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573-621. MR 30 #3992. MR 0173783 (30:3992)
  • [5] M. Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka J. Math. 4 (1967), 183-215. MR 37 #6999. MR 0231444 (37:6999)
  • [6] -, On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan 21 (1969), 58-93. MR 38 #5291. MR 0236998 (38:5291)
  • [7] -, Dirichlet spaces and their representations, Seminar on Probability 31 (1969). (Japanese)
  • [8] -, On Dirichlet spaces and Dirichlet rings, Proc. Japan Acad. 45 (1969), 433-436. MR 0253395 (40:6610)
  • [9] -, Dirichlet spaces and strong Markov processes (to appear).
  • [10] I. M. Gel'fand, D. A. Raĭkov and G. E. Šilov, Commutative normed rings, Fizmatgiz, Moscow, 1960; English transl., Chelsea, New York, 1964. MR 23 #A1242; MR 34 #4940.
  • [11] F. Knight, Note on regularization of Markov processes, Illinois J. Math. 9 (1965), 548-552. MR 31 #1713. MR 0177450 (31:1713)
  • [12] H. Kunita and T. Watanabe, Some theorems concerning resolvents over locally compact spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Prob. (Berkeley, Calif., 1965/66), vol. II: Contributions to Probability Theory, part 2, Univ. of California Press, Berkeley, Calif., 1967, pp. 131-164. MR 35 #4999. MR 0214148 (35:4999)
  • [13] L. H. Loomis, An introduction to abstract harmonic analysis, Van Nostrand, Princeton, N. J., 1953. MR 14, 883. MR 0054173 (14:883c)
  • [14] S. Mizohata, The theory of partial differential equations, Contemporary Math., no. 9, Iwanami Shoten, Tokyo, 1965. (Japanese) MR 38 #396. MR 0232070 (38:396)
  • [15] D. Ray, Resolvents, transition functions, and strongly Markovian processes, Ann. of Math. (2) 70 (1959), 43-72. MR 21 #6027. MR 0107302 (21:6027)
  • [16] T. Shiga and T. Watanabe, On Markov chains similar to the reflecting barrier Brownian motion, Osaka J. Math. 5 (1968), 1-33. MR 39 #7676. MR 0246372 (39:7676)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60.60

Retrieve articles in all journals with MSC: 60.60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0281256-1
Keywords: $ D$-space, underlying space, $ {L^2}$-resolvent, regular $ D$-space, Ray resolvent, strongly regular $ D$-space, Ray process, regular representation, continuous embedding
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society