Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Regular representations of Dirichlet spaces
HTML articles powered by AMS MathViewer

by Masatoshi Fukushima PDF
Trans. Amer. Math. Soc. 155 (1971), 455-473 Request permission


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.
  • A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208–215. MR 106365, DOI 10.1073/pnas.45.2.208
  • Jacques Deny, Principe complet du maximum et contractions, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 259–272 (French). MR 188475
  • J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 21 (1945), 305–370 (1955) (French). MR 74787
  • J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573–621. MR 173783
  • Masatoshi Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka Math. J. 4 (1967), 183–215. MR 231444
  • Masatoshi Fukushima, On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan 21 (1969), 58–93. MR 236998, DOI 10.2969/jmsj/02110058
  • —, Dirichlet spaces and their representations, Seminar on Probability 31 (1969). (Japanese)
  • Masatoshi Fukushima, On Dirichlet spaces and Dirichlet rings, Proc. Japan Acad. 45 (1969), 433–436. MR 253395
  • —, Dirichlet spaces and strong Markov processes (to appear). 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.
  • Frank Knight, Note on regularization of Markov processes, Illinois J. Math. 9 (1965), 548–552. MR 177450
  • Hiroshi Kunita and Takesi Watanabe, Some theorems concerning resolvents over locally compact spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 131–164. MR 0214148
  • Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
  • Sigeru Mizohata, Henbidun hôteisiki ron, Contemporary Mathematics, vol. 9, Iwanami Shoten, Tokyo, 1965 (Japanese). MR 0232070
  • Daniel Ray, Resolvents, transition functions, and strongly Markovian processes, Ann. of Math. (2) 70 (1959), 43–72. MR 107302, DOI 10.2307/1969891
  • Tokuzo Shiga and Takesi Watanabe, On Markov chains similar to the reflecting barrier Brownian motion, Osaka Math. J. 5 (1968), 1–33. MR 246372
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
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 155 (1971), 455-473
  • MSC: Primary 60.60
  • DOI:
  • MathSciNet review: 0281256