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

Vitali Liskevich; Michael Röckner

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1998)

  • Volume: 27, Issue: 1, page 69-91
  • ISSN: 0391-173X

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Liskevich, Vitali, and Röckner, Michael. "Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 27.1 (1998): 69-91. <http://eudml.org/doc/84355>.

@article{Liskevich1998,
author = {Liskevich, Vitali, Röckner, Michael},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Dirichlet forms; Dirichlet operators; strong uniqueness; Markov uniqueness},
language = {eng},
number = {1},
pages = {69-91},
publisher = {Scuola normale superiore},
title = {Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization},
url = {http://eudml.org/doc/84355},
volume = {27},
year = {1998},
}

TY - JOUR
AU - Liskevich, Vitali
AU - Röckner, Michael
TI - Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1998
PB - Scuola normale superiore
VL - 27
IS - 1
SP - 69
EP - 91
LA - eng
KW - Dirichlet forms; Dirichlet operators; strong uniqueness; Markov uniqueness
UR - http://eudml.org/doc/84355
ER -

References

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  1. [1] S. Albeverio - Y U.G. Kondratiev - M. Rockner, An approximate criterium of essential self-adjointness of Dirichlet operators, Potential Anal.1 (1992), 307-317. Zbl0808.47021MR1245233
  2. [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. Zbl0808.47022MR1246751
  3. [3] S. Albeverio - Y U.G. Kondratiev - M. Röckner, Dirichlet operators via stochastic analysis, J. Funct. Anal.128 (1995), 102-138. Zbl0820.60042MR1317712
  4. [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. Zbl0892.60008MR1472365
  5. [5] S. Albeverio - M. Röckner, Dirichlet forms on topological vector spaces-construction of an associated diffusion process, Probab. Theory Related Fields83 (1989), 405-434. Zbl0661.60094MR1017404
  6. [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: Longman1989. Zbl0691.60044MR1020060
  7. [7] S. Albeverio - M. Röckner, Classical Dirichletforms on topological spaces-Closability and Cameron-Martin formula, J. Funct. Anal.88 (1990), 395-436. Zbl0737.46036MR1038449
  8. [8] S. Albeverio - M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields89 (1991), 347-386. Zbl0725.60055MR1113223
  9. [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 Island1995. Zbl0824.31005MR1335494
  10. [10] S. Albeverio - M. Röckner - T.S. Zhang, Girsanov transform for symmetric diffusions with infinite dimensional state space., Ann. Probab.21 (1993), 961-978. Zbl0776.60093MR1217575
  11. [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. CanadaXV (1993), 1-6. Zbl0767.31009MR1214208
  12. [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. Zbl0827.31007MR1268239
  13. [13] Yu. M. Berezanskii - Yu. G. Kondratiev, "Spectral methods in infinite dimensional analysis", Naukova Dumka, Kiev, 1988. MR978630
  14. [14] V.I. Bogachev - N. Krylov - M. Röckner, Elliptic regularity and essential self-adjointness of Dirichlet operators on Rn, SFB-343 Preprint (1996). MR1391637
  15. [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. Zbl0657.60084MR967896
  16. [16] N. Bouleau - F. Hirsch, "Dirichlet Forms and Analysis on Wiener Space", de Gruyter, Berlin-New York, 1991. Zbl0748.60046MR1133391
  17. [17] E.B. Davies, "Heat Kernels and Spectral Theory", Cambridge University Press, Cambridge-New York- New Rochelle-Melbourne-Sydney, 1989. Zbl0699.35006MR990239
  18. [18] G. Da Prato - L. Tubaro, Introduction to stochastic quantization, Preprint (1996). Zbl0853.60052
  19. [19] A. Eberle, Uniqueness and non-uniqueness of singular diffusion operators, Doktorarbeit, Bielefeld (1997). Zbl0902.35063
  20. [20] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes", de Gruyter, Berlin- New York,1994. Zbl0838.31001MR1303354
  21. [21] D. Gatarek - B. Goldys, Existence, uniqueness and ergodicity for stochastic quantization equations, Preprint (1995). MR1391475
  22. [22] J. Glimm - A. Jaffe, "Quantum Physics: A Functional Integral Point of View", New York/Heidelberg/Berlin: Springer1996. Zbl0461.46051MR887102
  23. [23] Y.Z. Hu - G. Kallianpur, Exponential integrability and applications to stochastic quantization, Preprint (1996). Zbl0903.60046MR1610803
  24. [24] G. Jona-Lasinio - P.K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys.101 (1985), 406-436. Zbl0588.60054MR815192
  25. [25] G. Jona-Lasinio - P.K. Mitter, Large deviation estimates in the stochastic quantization of Φ42, Comm. Math. Phys.130 (1990), 111-121. Zbl0703.60095
  26. [26] G. Jona-Lasinio - R. Seneor, On a class of stochastic reaction-diffusion equations in two space dimensions, J. Phys.A24 (1991), 4123-4128. Zbl0745.60058MR1126653
  27. [27] N.V. Krylov, "Lectures on elliptic and parabolic equations in Hölder spaces", Graduate Studies in Mathematics, Vol. 12. American Mathematical Society, 1996. Zbl0865.35001MR1406091
  28. [28] J.L. Lions - E. Magenes, Non-homogeneous boundary value problems and applications, in: Grundlehren Math. Wiss., Berlin: Springer, 1972. Zbl0223.35039
  29. [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 equations11, pp. 163-217 (eds M.Demuth at al.) Akademie Verlag, Berlin, 1996. Zbl0854.47027MR1409835
  30. [30] V.A. Liskevich, Smoothness estimates and uniqueness for the Dirichlet operator, in: Operator Theory: Advances and Applications70 (1994), 149-152. Zbl0812.47051MR1309017
  31. [31] V.A. Liskevich - Yu. A. Semenov, Dirichlet operators: a priori estimates and the uniqueness problem, J. Funct. Anal.109 (1992), 199-213. Zbl0788.47041MR1183610
  32. [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. Zbl0826.31001MR1214375
  33. [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. Zbl0938.60047
  34. [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. MR853370
  35. [35] R. NAGEL (editor), "One-Parameter Semigroups of Positive Operators", Lecture Notes in mathematics1184, Springer-Verlag, Berlin-Heidelberg- New York, 1986. Zbl0585.47030MR839450
  36. [36] E. Nelson, The free Markov field, J. Funct. Anal.12 (1973), 211-227. Zbl0273.60079MR343816
  37. [37] G. Parisi - Y.S. Wu, Perturbation theory without gauge fixing, Scienta Sinica24 (1981), 383-496. MR626795
  38. [38] M. Reed - B. Simon, "Methods of Modem Mathematical Physics IV", Analysis of Operators. Orlando, FL: Academic Press, 1978. Zbl0401.47001
  39. [39] M. Röckner, Specifications and Martin boundaries for P (ϕ)2-random fields, Comm. Math. Phys.106 (1986), 105-135. Zbl0614.60068
  40. [40] M. Röckner - T.-S. Zhang, On uniqueness of generalized Schrödinger operators and applications, J. Funct. Anal.105 (1992), 187-231. Zbl0779.35028MR1156676
  41. [41] M. Röckner - T.-S. Zhang, Uniqueness of generalized Schrödinger operators and applications II, J. Funct. Anal.119 (1994), 455-467. Zbl0799.35053MR1261099
  42. [42] M. Röckner - T.S. Zhang, Finite dimensional approximation of diffusion processes on infinite dimensional state spaces, Stochastics Stochastics Rep.57 (1996), 37-55. Zbl0885.60066MR1407946
  43. [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. Zbl0855.31005MR1365253
  44. [44] B. Simon, "The P(ϕ)2-Euclidean (Quantum) Field Theory", Princeton, NJ: Princeton University Press, 1974. Zbl1175.81146
  45. [45] W. Stannat, (Non-symmetric) Dirichlet operators on L1: existence, uniquness and associated Markov processes, SFB-343-Bielefeld Preprint (1997). 
  46. [46] N. Wielens, On the essential self-adjointness of generalized Schrödinger operators, J. Funct. Anal.61 (1985), 98-115. Zbl0564.47010
  47. [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. Zbl0731.60005MR1022927

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