Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization

Liskevich, Vitali; Röckner, Michael

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 27 (1998) no. 1, pp. 69-91.

[1] S. Albeverio - YU.G. Kondratiev - M. Rockner, An approximate criterium of essential self-adjointness of Dirichlet operators, Potential Anal. 1 (1992), 307-317. | MR | Zbl

[2] S. Albeverio - Y.G. Kondratiev - M. Röckner, Addendum to: An approximate criterium of essential self-adjointness of Dirichlet operators, Potential Anal. 2 (1993), 195-198. | MR | Zbl

[3] S. Albeverio - YU.G. Kondratiev - M. Röckner, Dirichlet operators via stochastic analysis, J. Funct. Anal. 128 (1995), 102-138. | MR | Zbl

[4] S. Albeverio - Yu G. Kondratiev - M. Röckner, Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. SFB-343-Preprint 1996, To appear in: J. Funct. Anal. | MR | Zbl

[5] S. Albeverio - M. Röckner, Dirichlet forms on topological vector spaces-construction of an associated diffusion process, Probab. Theory Related Fields 83 (1989), 405-434. | MR | Zbl

[6] S. Albeverio - M. Röckner, "Dirichlet forms, quantum fields and stochastic quantization. Stochastic analysis, path integration and dynamics", Research Notes in Mathematics, vol. 200, 1-21. Editors: K. D. Elworthy, J. C. Zambrini. Harlow: Longman 1989. | MR | Zbl

[7] S. Albeverio - M. Röckner, Classical Dirichletforms on topological spaces-Closability and Cameron-Martin formula, J. Funct. Anal. 88 (1990), 395-436. | MR | Zbl

[8] S. Albeverio - M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89 (1991), 347-386. | MR | Zbl

[9] 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. | MR | Zbl

[10] S. Albeverio - M. Röckner - T.S. Zhang, Girsanov transform for symmetric diffusions with infinite dimensional state space., Ann. Probab. 21 (1993), 961-978. | MR | Zbl

[11] S. Albeverio - M. Röckner - T.S. Zhang, Markov uniqueness and its applications to martingale problems, stochastic differential equations and stochastic quantization, C.R. Math. Rep. Acad. Sci. Canada XV (1993), 1-6. | MR | Zbl

[12] S. Albeverio - M. Röckner - T.S. Zhang, Markov uniqueness for a class of infinite dimensional Dirichlet operators, in: Stochastic Processes and Optimal Control. Stochastic Monographs 7 (eds. H. J. Engelbert et al.), pp. 1-26, Gordon&Breach, 1993. | MR | Zbl

[13] Yu. M. Berezanskii - Yu. G. Kondratiev, "Spectral methods in infinite dimensional analysis", Naukova Dumka, Kiev, 1988. | MR

[14] V.I. Bogachev - N. Krylov - M. Röckner, Elliptic regularity and essential self-adjointness of Dirichlet operators on Rn, SFB-343 Preprint (1996). | MR

[15] V.S. Borkar - R.T. Chari - S.K. Mitter, Stochastic quantization offield theory in finite and infinite volume, J. Funct. Anal. 81 (1988), 184-206. | MR | Zbl

[16] N. Bouleau - F. Hirsch, "Dirichlet Forms and Analysis on Wiener Space", de Gruyter, Berlin-New York, 1991. | MR | Zbl

[17] E.B. Davies, "Heat Kernels and Spectral Theory", Cambridge University Press, Cambridge-New York- New Rochelle-Melbourne-Sydney, 1989. | MR | Zbl

[18] G. Da Prato - L. Tubaro, Introduction to stochastic quantization, Preprint (1996).

[19] A. Eberle, Uniqueness and non-uniqueness of singular diffusion operators, Doktorarbeit, Bielefeld (1997). | Zbl

[20] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes", de Gruyter, Berlin- New York,1994. | MR | Zbl

[21] D. Gatarek - B. Goldys, Existence, uniqueness and ergodicity for stochastic quantization equations, Preprint (1995). | MR

[22] J. Glimm - A. Jaffe, "Quantum Physics: A Functional Integral Point of View", New York/Heidelberg/Berlin: Springer 1996. | MR

[23] Y.Z. Hu - G. Kallianpur, Exponential integrability and applications to stochastic quantization, Preprint (1996). | MR

[24] G. Jona-Lasinio - P.K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101 (1985), 406-436. | MR | Zbl

[25] G. Jona-Lasinio - P.K. Mitter, Large deviation estimates in the stochastic quantization of Φ42, Comm. Math. Phys. 130 (1990), 111-121. | Zbl

[26] G. Jona-Lasinio - R. Seneor, On a class of stochastic reaction-diffusion equations in two space dimensions, J. Phys. A 24 (1991), 4123-4128. | MR | Zbl

[27] N.V. Krylov, "Lectures on elliptic and parabolic equations in Hölder spaces", Graduate Studies in Mathematics, Vol. 12. American Mathematical Society, 1996. | MR | Zbl

[28] J.L. Lions - E. Magenes, Non-homogeneous boundary value problems and applications, in: Grundlehren Math. Wiss., Berlin: Springer, 1972. | Zbl

[29] V.A. Liskevich - Yu. A. Semenov, Some problems on Markov semigroups, in: Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras Mathematical topics Advances in partial differential equations 11, pp. 163-217 (eds M.Demuth at al.) Akademie Verlag, Berlin, 1996. | MR | Zbl

[30] V.A. Liskevich, Smoothness estimates and uniqueness for the Dirichlet operator, in: Operator Theory: Advances and Applications 70 (1994), 149-152. | MR | Zbl

[31] V.A. Liskevich - Yu. A. Semenov, Dirichlet operators: a priori estimates and the uniqueness problem, J. Funct. Anal. 109 (1992), 199-213. | MR | Zbl

[32] Z.-M. Ma - M. Röckner, "Introduction to the Theory of (Non-symmetric) Dirichlet Forms, Springer-Verlag, Berlin-Heidelberg -New York- London- Paris-Tokyo, 1992. | MR | Zbl

[33] R. Mikulevicius - B.I. Rozowski, Martingale problems for stochastic PDE's. Preprint (1997), To appear in: Stochastic partial differential equations: Six Perspectives. Editors: R. Carmona, B. L. Rozowski. AMS Series Monographs and Reviews. | Zbl

[34] P.K. Mitter, Stochastic approach to Euclidean field theory (Stochastic quantization), in: New perspectives in Quantum Field Theory. Editors: J. Abad, M. Asorey, A. Cruz. Singapore: World Scientific, 1986. | MR

[35] R. NAGEL (editor), "One-Parameter Semigroups of Positive Operators", Lecture Notes in mathematics 1184, Springer-Verlag, Berlin-Heidelberg- New York, 1986. | MR | Zbl

[36] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973), 211-227. | MR | Zbl

[37] G. Parisi - Y.S. Wu, Perturbation theory without gauge fixing, Scienta Sinica 24 (1981), 383-496. | MR

[38] M. Reed - B. Simon, "Methods of Modem Mathematical Physics IV", Analysis of Operators. Orlando, FL: Academic Press, 1978. | Zbl

[39] M. Röckner, Specifications and Martin boundaries for P (ϕ)2-random fields, Comm. Math. Phys. 106 (1986), 105-135. | Zbl

[40] M. Röckner - T.-S. Zhang, On uniqueness of generalized Schrödinger operators and applications, J. Funct. Anal. 105 (1992), 187-231. | MR | Zbl

[41] M. Röckner - T.-S. Zhang, Uniqueness of generalized Schrödinger operators and applications II, J. Funct. Anal. 119 (1994), 455-467. | MR | Zbl

[42] M. Röckner - T.S. Zhang, Finite dimensional approximation of diffusion processes on infinite dimensional state spaces, Stochastics Stochastics Rep. 57 (1996), 37-55. | MR | Zbl

[43] I. Shigekawa, An example of regular (r, p)-capacity and essential self-adjointness of a diffusion operator in infinite dimensions, J. Math. Kyoto Univ. 35 (1995), 639-651. | MR | Zbl

[44] B. Simon, "The P(ϕ)2-Euclidean (Quantum) Field Theory", Princeton, NJ: Princeton University Press, 1974. | Zbl

[45] W. Stannat, (Non-symmetric) Dirichlet operators on L1: existence, uniquness and associated Markov processes, SFB-343-Bielefeld Preprint (1997).

[46] N. Wielens, On the essential self-adjointness of generalized Schrödinger operators, J. Funct. Anal. 61 (1985), 98-115. | Zbl

[47] J.A. Yan, "Generalizations of Gross' and Minlos' theorems", in: Séminaire de Probabilités XXII, (J. Azema et. al.), 395-404, Lect. Notes in Math. 1372. Berlin: Springer, 1989. | Numdam | MR | Zbl