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Lp-analysis of finite and infinite dimensional diffusion operators

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Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions

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References

  • [ABR 97] S. Albeverio, V.I. Bogachev, M. Röckner: On uniqueness of invariant measures for finite and infinite dimensional diffusions. SFB-343-Preprint 1997. To appear in: Commun. Pure and Appl Math., 46 Seiten.

    Google Scholar 

  • [AKR 96] S. Albeverio, Y.G. Kondratiev, M. Röckner: Dirichlet operators via stochastic analysis. J. Funct. Anal. 128, 102–138 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [AKR 97a] S. Albeverio, Y.G. Kondratiev, M. Röckner: Ergodicity of L 2 -semigroups and extremality of Gibbs states. J. Funct. Anal. 144, 394–423 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [AKR 98] S. Albeverio, Y.G. Kondratiev, M. Röckner: Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. J. Funct. Anal. 149, 415–469 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [AKR 98a] S. Albeverio, Y.G. Kondratiev, M. Röckner: Analysis and Geometry on configuration spaces. J. Funct. Anal. 154, 444–500 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [AKR 98b] S. Albeverio, Y.G. Kondratiev, M. Röckner: Analysis and Geometry on configuration spaces. The Gibbsian case. J. Funct. Anal. 157, 242–291 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [AKRT 98] S. Albeverio, Y.G. Kondratiev, M. Röckner, T. Tsikalenko: Existence and exponential moment bounds for symmetrizing measures and applications to Gibbs states. Preprint (1998), Publication in preparation.

    Google Scholar 

  • [AR 90] S. Albeverio, M. Röckner Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula. J. Funct. Anal. 88, 395–436 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  • [AR 91] S. Albeverio, M. Röckner Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Th. Rel. Fields 89, 347–386 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [AR 95] S. Albeverio, M. Röckner Dirichlet form methods for uniqueness of martingale problems and applications In: Stochastic Analysis. Proceedings of Symposia in Pure Mathematics Vol. 57, 513–528. Editors: M.C. Cranston, M.A. Pinsky. Am. Math. Soc.: Providence, Rhode Island 1995.

    Chapter  Google Scholar 

  • [Ar 86] W. Arendt: The abstract Cauchy problem, special semigroups and perturbation In: One-parameter semigroups of positive operators, Edited by R. Nagel, Berlin: Springer 1986.

    Chapter  Google Scholar 

  • [BDPR 96] V.I. Bogachev, G. Da Prato, M. Röckner: Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators, NoDEA 3, 261–268 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [BKR 96] V.I. Bogachev, N. Krylov, M. Röckner: Regularity of invariant measures: the case of non-constant diffusion part.. J. Funct. Anal. 138, 223–242 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [KBR 97] V.I. Bogachev, N. Krylov, M. Röckner: Elliptic regularity and essential self-adjointness of Dirichlet operators on R d Ann. Scuola Norm. Sup. Pisa. Cl. Sci., Serie IV, Vol. XXIV. Fasc. 3, 451–461 (1997).

    MathSciNet  MATH  Google Scholar 

  • [BR 95] V.I. Bogachev, M. Röckner: Regularity of invariant measures on finite and infinite dimensional spaces and applications J. Funct. Anal. 133, 168–223 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [BR 95a] V.I. Bogachev, M. Röckner: Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift. Osaka J. Math. 32, 237–274 (1995).

    MathSciNet  MATH  Google Scholar 

  • [BR 98] V. I. Bogachev, M. Röckner: A generalization of Hasminski’s theorem on existence of invariant measures for locally integrable drifts. SFB-343-Preprint 1998. To appear in: Theory Prob. Appl., 18 Seiten.

    Google Scholar 

  • [BRS 96] V.I. Bogachev, M. Röckner, B. Schmuland: Generalized Mehler semigroups and applications. Probab. Th. Rel. Fields 105, 193–225 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [BRZ 97] V.I. Bogachev, M. Röckner, T.S. Zhang: Existence and uniqueness of invariant measures: an approach via sectorial forms. SFB-343-Preprint 1997. To appear in: Appl. Math. Optim. 28 Seiten.

    Google Scholar 

  • [Dav 85] E.B. Davies: L 1 -Properties of second order elliptic operators. Bull London Math. Soc. 17, 417–436 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [Dav 89] E.B. Davies: Heat kernels and spectral theory. Cambridge University Press 1989.

    Google Scholar 

  • [Eb 97] A. Eberle: Uniqueness and non-uniqueness of singular diffusion operators. Doctor-Thesis, Bielefeld 1997, SFB-343-Preprint (1998), 291 pages, publication in preparation.

    Google Scholar 

  • [EthKur 86] S.N. Ethier, T.G. Kurtz: Markov processes. Characterization and convergence. New York: Wiley 1986.

    Book  MATH  Google Scholar 

  • [FR 97] M. Fuhrman, M. Röckner: Generalized Mehler semigroups: The non-Gaussian case. FSP-Universität Bielefeld-Preprint 1997, 37 Seiten. To appear in: Potential Analysis.

    Google Scholar 

  • [Gr 93] L. Gross: Logarithmic Sobolev inequalities and contractive properties of semigroups. Lect. Notes Math. 1563, 54–82. Berlin: Springer 1993.

    MATH  Google Scholar 

  • [Ko 37] A.N. Kolmogorov: Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113, 766–772 (1937).

    Article  MathSciNet  MATH  Google Scholar 

  • [L 98] V. Liskevich: On the uniqueness problem for Dirichlet operators. Preprint 1998.

    Google Scholar 

  • [LR 97] V. Liskevich, M. Röckner: Strong uniqueness for a class of infinite dimensional Dirichlet operators and applications to stochastic quantization. SFB-343-Preprint 1997. To appear in: Ann. Scuola Norm. di Pisa, 25 Seiten.

    Google Scholar 

  • [MOR 95] L. Overbeck, Z.M. Ma, M. Röckner: Markov processes associated with Semi-Dirichlet forms. Osaka J. Math. 32, 97–119 (1995).

    MathSciNet  MATH  Google Scholar 

  • [MR 92] Z.M. Ma, M. Röckner: An introduction to the theory of (non-symmetric) Dirichlet forms. Berlin: Springer 1992.

    Book  MATH  Google Scholar 

  • [MR 95] Z.M. Ma, M. Röckner: Markov processes associated with positivity preserving forms. Can. J. Math. 47, 817–840 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [Pa 85] A. Pazy: Semigroups of linear operators and applications to partial differential equations. Berlin: Springer 1985.

    MATH  Google Scholar 

  • [ReS 75] M. Reed, B. Simon: Methods of modern mathematical physics II. Fourier Analysis. New York-San Francisco-London: Academic Press 1975.

    MATH  Google Scholar 

  • [R 98] M. Röckner: Stochastic analysis on configuration spaces: basic ideas and recent results. In: New directions in Dirichlet forms, 157–231. Editors: J. Jost et al. Studies in Advanced Mathematics, International Press, 1998.

    Google Scholar 

  • [Sh 98] I. Shigekawa: A non-symmetric diffusion process on the Wiener space. Preprint 1998.

    Google Scholar 

  • [St 96] W. Stannat: The theory of generalized Dirichlet forms and its applications in analysis and stochastics. Doctor-Thesis, Bielefeld 1996, SFB-343-Preprint (1996), 100 pages. To appear in: Memoirs of the AMS.

    Google Scholar 

  • [St 97] W. Stannat: (Nonsymmetric) Dirichlet operators on L 1 : existence, uniqueness and associated Markov processes. SFB-343-Preprint (1997), 38 pages. Publication in preparation.

    Google Scholar 

  • [Tr 73] N.S. Trudinger: Linear elliptic operators with measurable coefficients. Ann. Scuola Normale Sup. Pisa 27, 265–308 (1973).

    MathSciNet  MATH  Google Scholar 

  • [T 97] G. Trutnau: Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions. SFB-343-Preprint (1998), 29 pages. Publication in preparation.

    Google Scholar 

  • [T 98] G. Trutnau: Doctor-Thesis, Bielefeld, in preparation.

    Google Scholar 

  • [Ya 89] J.A. Yan: Generalizations of Gross’ and Minlos’ theorems. In: Azema, J., Meyer, P.A., Yor, M. (eds.) Séminaire de Probabilités. XXII (Lect. Notes Math., vol. 1372, pp. 395–404) Berlin-Heidelberg-New York: Springer 1989.

    Chapter  Google Scholar 

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Röckner, M. (1999). Lp-analysis of finite and infinite dimensional diffusion operators. In: Da Prato, G. (eds) Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Mathematics, vol 1715. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092418

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