Fractional order Sobolev spaces on Wiener space

Springer Science and Business Media LLC - Tập 95 - Trang 175-198 - 1993
Shinzo Watanabe1
1Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, Japan

Tóm tắt

Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Hölder continuity of a class of conditional expectations is obtained.

Tài liệu tham khảo

Adams, R.A.: Sobolev spaces. New York: Academic Press 1978 Airault, H., Malliavin, P.: Intégration géométrique sur l'espace de Wiener. Bull. Sci. Math.112, 3–52 (1988) Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: de Gruyter 1991 Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. 2nd edn. Amsterdam New York: North-Holland/Kodansha 1988 Kuo, H.-H.: Donsker's delta function as a generalized Brownian functionals and its application. In: Theory and applications of random fields. Proc. IFIP Conf. Bangalore 1982. Kallianpur, G. (ed.),LNCI 49, 167–178, (1983) Kusuoka, S.: On the foundation of Wiener-Riemannian manifolds. In: Elworthy, K.D., Zambrini, J.-C. (eds.) Stochastic analysis, path integration and dynamics, pp. 130–164 Harlow, Essex: Longman 1989 Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus, I. In: Ito, K. (ed.), Stochastic analysis, pp. 271–306. Proc. Taniguchi Symp. Katata and Kyoto 1982. Tokyo: Kinokuniya 1984 Lions, J.-L.: Sur les espaces d'interpolation; dualit'e, Math. Scand.9, 147–177 (1961) Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Ito K. (ed.), Stochastic analysis, pp. 369–386. Proc. Taniguchi Symp. Katata and Kyoto 1982. Tokyo: Kinokuniya 1984 Meyer, P.A.: Retour sur la théorie de Littlewood-Paley. In: Azéma, J., Yor, M. (eds.), Séminaire de Prob. XV, 1979/1980LNM 850, 151–166 (1981) Shigekawa, I.: Derivatives of Wiener functionals, and absolute continuity of induced measures. J. Math. Kyoto Univ.20, 263–289 (1980) Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, N.J.: Princeton University Press 1970 Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton, N.J.: Princeton University Press 1971 Sugita, H.: Sobolev space of Wiener functionals and Malliavin calculus. J. Math. Kyoto Univ.25, 31–48 (1985) Sugita, H.: On a characterization of the Sobolev spaces over an abstract Wiener space. J. Math. Kyoto Univ.25, 717–757 (1985) Sugita, H.: Positive generalized Wiener functionals and potential theory over abstract Wiener spaces. Osaka J. Math.25, 665–698 (1988) Watanabe, S.: Malliavin's calculus in terms of generalized Wiener functionals. In: Kalliapur, G. (ed.) Theory and application of random fields. Proc. IFIP Conf. Bangalore 1982,LNCI 49, 284–290 (1983) Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984 Watanabe, S.: Donsker's δ-functions in the Malliavin calculus. In: Mayer-Wolf, E., Merzbach, E., Shwartz, A. (eds.) Stochastic analysis, liber amicorum for Moshe Zakai, pp. 495–502. New York: Academic Press 1991 Watanabe, S.: Some refinement of conditional expectations on Wiener space by means of the Malliavin calculus. Proc. 6-th USSR-Japan Symp. on Probability Theory, pp. 414–421. Singapore: World Scientific 1992