Potential Analysis

, Volume 3, Issue 1, pp 159–170 | Cite as

Some calculations on the conditional densities of well-admissible measures on linear spaces

  • Shiqi Song


When a probability measurem on a topological vector spaceE is well-admissible in a directionk εE, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.

Mathematics Subject Classification (1991)


Key words

Dirichlet form regularity of a Dirichlet form capacity of a Dirichlet form topological vector space admissible measure admissible vector 


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  1. 1.
    Albeverio, S. and Hoegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces,Z. W. verw. Gebiete 40 (1977), 1–57.Google Scholar
  2. 2.
    Albeverio, S. and Kusuoka, S.: Maximality of infinite dimensional Dirichlet forms and Hoegh-Krohn's model of quantum fields,Ideas and Methods in Quantum and Statistical Physics, Vol. 2. Edited by Albeverio-Fenstad-Holden-Lindstrom, Cambridge University Press, 1992.Google Scholar
  3. 3.
    Albeverio, S., Kusuoka, S., and Röckner, M.: On partial integration in infinite dimensional space and applications to Dirichlet forms,J. London Math. Soc. 42 (1990), 122–136.Google Scholar
  4. 4.
    Albeverio, S. and Röckner, M.: Classical Dirichlet forms on topological vector spaces — closability and a Cameron-Martin formula,J. Func. Anal. 88 (1990), 395–436.Google Scholar
  5. 5.
    Albeverio, S. and Röckner, M.: Stochastic differential equations in infinite dimensions: solution via Dirichlet forms. Preprint Edinburgh, 1989.Google Scholar
  6. 6.
    Fukushima, M.:Dirichlet Forms and Markov Processes, Amsterdam-Oxford-New York: North Holland, 1980.Google Scholar
  7. 7.
    Fukushima, M.:On Absolute Continuity of Multidimensional Symmetrizable Diffusions, in: Lect. Notes in Math.923, 146–176. Berlin-Heidelberg-New York: Springer, 1982.Google Scholar
  8. 8.
    Ma, Z. M. and Röckner, M.:An Introduction to the Theory of (Non Symmetric) Dirichlet Forms. Book in preparation, 1991.Google Scholar
  9. 9.
    Röckner, M.: Lecture on Dirichlet forms on infinite dimensional state space and applications, 1990.Google Scholar
  10. 10.
    Röckner, M. and Song, S. Q.: Well admissibility, change of probability and conditional density. Preprint 1991.Google Scholar
  11. 11.
    Röckner, M. and Zheng, T. S.: Uniqueness of generalized Schrödinger operators and applications. Preprint 1991.Google Scholar
  12. 12.
    Schmuland, B.: An alternative compactification for classical Dirichlet forms on topological vector spaces,Stochastics 33, 1990, 75–90.Google Scholar
  13. 13.
    Song, S.: A study on Markovian maximality, change of probability and regularity, to appear inPotential Analysis.Google Scholar
  14. 14.
    Song, S. Q.: Differentiability of the conditional densities of well-admissible measures on linear spaces. Preprint 1991.Google Scholar
  15. 15.
    Song, S.: Admissible vectors and their associated Dirichlet forms,Potential Analysis,1 (1992), 319–336.Google Scholar
  16. 16.
    Song, S.: Compactifications of Banach spaces and construction of Diffusion processes,Acta Math. Appl. (in print).Google Scholar
  17. 17.
    Takeda, M.: On the uniqueness of Markovian self-adjoint extension of diffusion operators on infinite dimensional space,Osaka J. Math. 22 (1985), 733–742.Google Scholar
  18. 18.
    Takeda, M.:On the Uniqueness of Markovian Self-Adjoint Extension, Lect. Notes in Math.1250 (Stochastic processes — Mathematics and Physics), 319–325, Berlin, Springer, 1985.Google Scholar
  19. 19.
    Takeda, M.: The maximum Markovian self-adjoint extensions of generalized Schrödinger operators. Preprint 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Shiqi Song
    • 1
  1. 1.Equipe d'Analyse et ProbabilitésUniversité Evry Val d'EssonneEvry CedexFrance

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