References
S. Albeverio, Yu. G. Kondratiev, and M. Röckner, “Dirichlet operators via stochastic analysis,”J. Funct. Anal.,128, 102–138 (1995).
A. V. Antoniouk and A. V. Antoniouk, “Smoothing properties of semigroups for Dirichlet operators of Gibbs measures,”J. Funct. Anal.,127, 390–430 (1995).
A. Beck, “Uniqueness of flow solutions of differential equations,” In:Lect. Notes Math., Vol. 318 (1973), pp. 30–50.
D. Bell, “A quasi-invariance theorem for measures on Banach spaces,”Trans. Amer. Mat. Soc.,290, No. 2, 841–845 (1985).
D. Bell, “The Malliavin calculus,” In:Pitman Monogr. Surv. Pure Appl. Math., Vol. 34, Wiley, New York (1987).
D. Bell, “Transformations of measure on an infinite-dimensional vector space,” In:Seminar on Stochastic Processes, 1990 (Vancouver, BC, 1990), Progr. Probab., Vol. 24, Birkhäuser Boston, Boston, MA (1991).
V. I. Bogachev, “Locally convex spaces with the CLT property and supports of measures,”Moscow Univ. Math. Bull.,41, No. 6, 19–23 (1986).
V. I. Bogachev,Differentiable measures and the Malliavin calculus, Preprint No. 16 (1995), Sc. Norm. Super., Pisa (1995).
V. I. Bogachev, “Deterministic and stochastic differential equations in infinite dimensional spaces,”Acta Appl. Math.,40, 25–93 (1995).
V. I. Bogachev,Gaussian Measures [in Russian], Fizmatlit, Moscow (1997).
V. I. Bogachev and E. Mayer-Wolf,Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions, Preprint 96-003 SFB 343 Univ. Bielefeld (1996).
V. I. Bogachev and E. Mayer-Wolf, “Flows generated by Sobolev type vector fields and the corresponding transformations of probability measures,”Dokl. Rossiisk. Akad. Nauk (in press).
V. I. Bogachev and M. Röckner, “Regularity of invariant measures on finite and infinite dimensional spaces and applications,”J. Funct. Anal.,133, 168–223 (1995).
V. I. Bogachev, M. Röckner, and B. Schmuland,Generalized Mehler semigroups and applications, Preprint SFB 343 No. 94-088, Bielefeld Univ. (1994); to appear inProbab. Theor. Relat. Fields.
V. I. Bogachev and O. G. Smolyanov, “Analytic properties of infinite dimensional distributions”Russian Math. Surv.,45, No. 3, 3–104 (1990).
Ch. Borell, “Gaussian Radon measures on locally convex spaces,”Math. Scand.,38, No. 2, 265–284 (1976).
A. Brandao, “Global finite dimensional flows,”Expos. Math.,13, 377–384 (1995).
R. Buckdahn, “Anticipative Girsanov transformations”Probab. Theor. Relat. Fields,89, 211–238 (1991).
A.-B. Cruzeiro, “Équations différentielles ordinaires: Non explosion et mesures quasi-invariantes,”J. Funct. Anal.,54, 193–205 (1983).
A.-B. Cruzeiro, “Équations différentielles sur l'espace de Wiener et formules de Cameron-Martin non linéaires,”J. Funct. Anal.,54, 206–227 (1983).
A.-B. Cruzeiro, “Unicité de solutions d'équations différentielles sur l'espace de Wiener,”J. Funct. Anal.,58, 335–347 (1984).
A.-B. Cruzeiro, “Estimations capacitaires sur l'espace de Wiener, I,”Bull. Sci. Math.,110, 139–147 (1986).
A.-B. Cruzeiro, “Flows in infinite dimensions and associated transformations of Gaussian measures,” In:Stochastic methods in mathematics and physics (Karpacz, 1988), World Sci. Publishing, Teaneck, NJ (1989), pp. 290–301.
A.-B. Cruzeiro and P. Malliavin, “Repère mobile et géometrie riemannienne sur les espaces des chemins,”C. R. Acad. Sci. Paris, Sér. 1,319, 859–864 (1994).
A.-B. Cruzeiro and P. Malliavin,Curvatures of path spaces and stochastic analysis, Report No. 16, Institut Mittag-Leffler (1995).
Yu. L. Daletskii and G. Sohadze, “Absólute continuity of smooth measures,”Funct. Anal. Appl.,22, No. 2, 77–78 (1988).
Yu.L. Daletskii and S. V. Fomin,Measures and Differential Equations in Infinite-Dimensional Spaces, Kluwer Acad. Publ. (1991).
R. J. DiPerna and P. L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces,”Invent. Math.,98, No. 3, 511–547 (1989).
R. J. DiPerna and P. L. Lions, “Équations differentielles ordinaires et équations de transport avec des coefficients irréguliers,” In:Séminaire EDP 1988–1989, Ecole Polytechnique, Palaiseau (1989).
B. Driver, “A Cameron-Martin type quasi-invariance theorem for the Brownian motion on a compact manifold,”J. Funct. Anal.,110, 272–376 (1992).
S. Kusuoka, “The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity,”J. Fac. Sci. Univ. Tokyo, Sect. IA,29, 567–597 (1982).
J. A. Leon and Ph. Protter, “Some formulas for anticipative Girsanov transformations,” In:Chaos Expansions, Multiple Wiener-Ito Integrals and Their Applications, Guanjuato (1992), pp. 267–291.
M. Malliavin and P. Malliavin, “Integration on loop groups, I. Quasi-invariant measures,”J. Funct. Anal.,93, 207–237 (1990).
P. Malliavin,Géometrie Différentielle Stochastique, Univ. de Montreal Presses (1978).
P. Malliavin, “Smooth σ-fields,” In:Stochastic Analysis, Academic Press, New York-Boston (1991), pp. 371–382.
P. Malliavin, “Infinite dimensional analysis,”Bull. Sci. Math.,117, 63–90 (1993).
E. Mayer-Wolf, “Preservation of measure continuity under conditioning,”J. Funct. Anal.,115, 227–246 (1993).
D. Nualart, A. S. Ustunel, and M. Zakai, “Some relations among classes of σ-fields,”Probab. Theor. Relat. Fields,85, 119–129 (1990).
G. Peters, “Flows on the Wiener space generated by vector fields with low regularity,”C. R. Acad. Sci. Sér. 1,320, 1003–1008 (1995).
G. Peters, “Anticipating flows on the Wiener space generated by vector fields of low regularity,”J. Funct. Anal.,142, 129–192 (1996).
G. Pisier, “Probability methods in the geometry of Banach spaces,” In:Lect. Notes Math., Vol. 1206 (1985), pp. 167–241.
R. Ramer, “On nonlinear transformations of Gaussian measures,”J. Funct. Anal.,15, 166–187 (1974).
A. V. Skorohod,Integration in Hilbert Space, Springer (1974).
O. G. Smolyanov and H. Weizsäcker, “Differentiable families of measures,”J. Funct. Anal.,118, 454–476 (1993).
A. S. Üstünel and M. Zakai, “On the structure of independence on Wiener space,”J. Funct. Anal.,90, No. 1, 113–137 (1990).
A. S. Üstünel and M. Zakai, “Transformation of Wiener measure under anticipative flows,”Probab. Theor. Relat. Fields,93, 91–136 (1992).
A. S. Üstünel and M. Zakai, “Analyse de rotations aléatoires sur l'espace de Wiener,”C. R. Acad. Sci. Sér. 1,319, 1069–1073 (1994).
A. S. Üstünel and M. Zakai, “Random rotations of the Wiener path,”Probab. Theor. Relat. Fields,103, No. 3, 409–430 (1995).
N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan,Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).
M. Zakai and O. Zeitouni, “When does the Ramer formula look like the Girsanov formula?”Ann. Probab.,20, No. 3, 1436–1440 (1992).
W. Ziemer,Weakly Differentiable Functions, Springer (1989).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 42, Dinamicheskie Sistemy-6, 1997.
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Bogachev, V., Mayer-Wolf, E. Dynamical systems generated by sobolev class vector fields in finite and infinite dimensions. J Math Sci 94, 1394–1445 (1999). https://doi.org/10.1007/BF02365019
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DOI: https://doi.org/10.1007/BF02365019