Abstract
We use a stochastic integral which was first constructed by Nualart, Zakai and Ogawa, to show, for the variables of the second Wiener chaos, that the existence of this integral imply that these variables possess an approximate limit with respect to measurable norms defined by Gross. Moreover, this limit does not depend on the choice of the norm. Furthermore, we show that measurable norms possess an approximate limit with respect to quadratic norms. The main argument is a correlation inequality proved by Hargé.
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References
Balakrishnan, A. V.: Applied Functional Analysis, Springer-Verlag, New York, 1963.
Ben Arous, G., Gradinaru, M. and Ledoux, M.: 'Hölder norms and the support theorem for diffusions', Ann. Inst. H. Poincaré Probab. Statist. 30(3) (1994), 415-436.
Bogachev, V. I.: 'The Onsager-Machlup functions for Gaussian measures', Dokl. Math. 52(2) (1995), 216-218.
Borell, C.: 'A note on Gauss measures which agree on small balls', Ann. Inst. H. Poincaré 13 (1977), 231-238.
Bouleau, N. and Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space, de Gruyter Studies in Math. 14, 1991.
Carmona, R. A. and Nualart, D.: 'Traces of random variables on Wiener space and Onsager-Machlup functional', J. Funct. Anal. 107(2) (1992), 402-438.
Fang, S.: 'Sur la continuité approximative forte des fonctionnelles d'Itô', Stochastics Stochastics Rep. 36 (1991), 193-204.
Gross, L.: 'Measurable functions on Hilbert space', Trans. Amer. Math. Soc. 105 (1962), 372-390.
Gross, L.: 'Abstract Wiener spaces', in: Proceedings of the Fifth Berkeley Symp. Math. Statist. Prob. II, Part I, 1967, pp. 31-42.
Hargé, G.: 'Continuité approximative de certaines fonctionnelles sur l'espace de Wiener', J. Math. Pures Appl. 74 (1995), 59-93.
Hargé, G.: 'Une inégalité de décorrelation pour la mesure gaussienne', C.R. Acad. Sci. Paris 326 (1998), 1325-1328. À paraitre dans Ann. Probab.
Hille, E. and Tamarkin, J. D.: 'On the characteristic values of linear integral equations', Acta Math. 57 (1931), 1-76.
Kuo, H. H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, 1975.
Ledoux, M.: Isoperimetry and Gaussian analysis', in: Ecole d'été de probabilités de Saint-Flour XXIV-1994, Lecture Notes in Math. 1648, Springer, 1996.
Ma, Z. M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, 1992.
Millet, A. and Nualart, D.: 'Support theorems for a class of anticipating stochastic differential equations', Stochastics Stochastics Rep. 39 (1992), 1-24.
Nualart, D.: The Malliavin Calculus and Related Topics, Springer-Verlag, New York, 1995.
Nualart, D. and Zakai, M.: 'Generalized stochastic integrals and the Malliavin calculus', Probab. Theory Related Fields 73 (1986), 255-280.
Ogawa, S.: 'Sur le produit direct du bruit blanc par lui-même', C.R. Acad. Sci. Paris Série A 288 (1979), 359-362.
Ogawa, S.: 'Quelques propriétés de l'intégrale stochastique du type non causal', Japan J. Appl. Math. 1 (1984), 405-416.
Shepp, L. A. and Zeitouni, O.: 'A note on conditional exponential moments and Onsager Machlup functionals', Ann. Probab. 20 (1992), 652-654.
Stroock, D. W. and Varadhan, S. R. S.: 'On the support of diffusion processes with applications to the strong maximum principal', in: Sixth Berkeley Symp. Math. Statist. Prob. 1972, pp. 333-359.
Sugita, H.: 'Various topologies in theWiener space and Lévy's stochastic area', Probab. Theory Related Fields 91(3/4) (1992), 283-296.
Mayer-Wolf, E. and Zeitouni, O.: 'Onsager Machlup functionals for non trace class SPDE'S', Probab. Theory Related Fields 95 (1993), 199-216.
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Hargé, G. Limites approximatives sur l'espace de Wiener. Potential Analysis 16, 169–191 (2002). https://doi.org/10.1023/A:1012629125419
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DOI: https://doi.org/10.1023/A:1012629125419