Summary
In this paper we study conditions ensuring that the law of aC([0, 1])-valued functional defined on an abstract Wiener space is absolutely continuous with respect to the Wiener measure onC([0,1]). These conditions extend those established byP. Malliavin [12, 13] for finite-dimensional Wiener functionals, and those of [15] for Hilbert-valued functionals.
Article PDF
Similar content being viewed by others
References
Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 468–505 (1981)
Bouleau, N., Hirsch, F.: Propriétés d'absolue continuité dans les espaces de Dirichlet et applications aux E.D.S. Séminaire de Probabilités XX. (Lect. Notes Math., vol. 1204, pp. 131–161) Berlin Heidelberg New York: Springer 1986
Doss, H.: Quelques propriétés des processus de diffusion à valeurs dans un espace de Hilbert. Thèse de 3ème cycle, Université Paris 6 1975
Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math.97, 1061–1083 (1976)
Hayes, C.A., Pauc, C.Y.: Derivation and martingales. Berlin Heidelberg New York: Springer 1970
Holley, R., Stroock, D.: Diffusions on an infinite dimensional torus. J. Funct. Anal.42, 29–63 (1981)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam New York: North Holland 1984
Ikeda, N., Watanabe, S.: An introduction to Malliavin's calculus. Ito, K. (ed.). Taniguchi Symp. Stoch. Anal., Katata 1982. Amsterdam New York: North Holland 1984
Korezlioglu, H., Mazziotto, G.: Sur les diffusions multidirectionnelles. C.R. Acad. Sci. Paris, Sér, A290, 657–660 (1980)
Kuo, H.H.: Gaussian measures on Banach spaces. (Lect. Notes Math., vol. 463) Berlin Heidelberg New York: Springer 1975
Kusuoka, S.: The nonlinear transformation of a Gaussian measure on Banach space and its absolute continuity (I). J. Fac. Sci., Univ. Tokyo, Sect. IA29, 567–597 (1982)
Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International conference on Stoch. Diff. Eq., Kyoto 1976, pp. 195–263, Kinokuta 1978
Malliavin, P.:C k-hypoellipticity with degeneracy. In: Friedman, A., Pinsky, M. (ed.) Stochastic analysis, pp. 199–214 and 327–340. New York London: Academic Press 1978
Meyer, P.A.: Quelques résultats analytiques sur le semi-groupe d'Ornstein-Uhlenbeck en dimension infinie. In: Kallianpur, G. (ed.) Th. and Appl. of Random fields. Proc. IFIP-WG 7/1, Bangalore. (Lect. Notes Control Inf. Sci., vol. 49, pp. 201–214) Berlin Heidelberg New York: Springer 1983
Moulinier, J.M.: Absolue continuité de probabilités de transition par rapport à une mesure Gaussiene dans un espace de Hilbert. J. Funct. Anal.64, 275–295 (1985)
Neveu, J.: Martingales à temps discret. Paris: Masson 1972
Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields78, 535–581 (1988)
Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin calculus. Probab. Th. Rel. Fields73, 255–280 (1986)
Ramer, R.: On nonlinear transformations of Gaussian measures. J. Funct. Anal.155, 166–187 (1974)
Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ.20, 263–289 (1980)
Stroock, D.: The Malliavin calculus and its applications to second order parabolic differential equations. Math. Syst. Theory14, 263–289 (1981)
Watanabe, S.: Stochastic differential equations and Malliavin calculus. Bombay: Tata Institute Publ. 1984
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mazziotto, G., Millet, A. Absolute continuity of the law of an infinite dimensional Wiener functional with respect to the Wiener probability. Probab. Th. Rel. Fields 85, 403–411 (1990). https://doi.org/10.1007/BF01193945
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01193945