Stochastic Poisson-Sigma model

Rémi Léandre[1]

  • [1] Institut de Mathématiques Faculté des Sciences Université de Bourgogne 21000 Dijon, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 4, page 653-667
  • ISSN: 0391-173X

Abstract

top
We produce a stochastic regularization of the Poisson-Sigma model of Cattaneo-Felder, which is an analogue regularization of Klauder’s stochastic regularization of the hamiltonian path integral [23] in field theory. We perform also semi-classical limits.

How to cite

top

Léandre, Rémi. "Stochastic Poisson-Sigma model." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.4 (2005): 653-667. <http://eudml.org/doc/84575>.

@article{Léandre2005,
abstract = {We produce a stochastic regularization of the Poisson-Sigma model of Cattaneo-Felder, which is an analogue regularization of Klauder’s stochastic regularization of the hamiltonian path integral [23] in field theory. We perform also semi-classical limits.},
affiliation = {Institut de Mathématiques Faculté des Sciences Université de Bourgogne 21000 Dijon, France},
author = {Léandre, Rémi},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {653-667},
publisher = {Scuola Normale Superiore, Pisa},
title = {Stochastic Poisson-Sigma model},
url = {http://eudml.org/doc/84575},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Léandre, Rémi
TI - Stochastic Poisson-Sigma model
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 4
SP - 653
EP - 667
AB - We produce a stochastic regularization of the Poisson-Sigma model of Cattaneo-Felder, which is an analogue regularization of Klauder’s stochastic regularization of the hamiltonian path integral [23] in field theory. We perform also semi-classical limits.
LA - eng
UR - http://eudml.org/doc/84575
ER -

References

top
  1. [1] H. Airault and P. Malliavin, “Quasi-sure Analysis”, Publication Université Paris VI, Paris, 1990. 
  2. [2] H. Airault and P. Malliavin, “Integration on Loop Groups”, Publication Université Paris VI, Paris, 1990. Zbl0787.22021
  3. [3] S. Albeverio, Loop groups, random gauge fields, Chern-Simons models, strings: some recent mathematical developments, In: “Espaces de Lacets”, R. Léandre, S. Paycha and T. Wurzbacher (eds.), Publi. Univ. Strasbourg, Strasbourg, 1996, 5–34. 
  4. [4] S. Albeverio, R. Léandre and M. Röckner, Construction of a rotational invariant diffusion on the free loop space. C.R. Acad. Sci. Paris Sér. I Math. 316 (1993), 287-292. Zbl0776.58041MR1205201
  5. [5] M. Arnaudon and S. Paycha, Stochastic tools on Hilbert manifolds: interplay with geometry and physics, Comm. Math. Phys. 197 (1997), 243–260. Zbl0888.58004MR1463828
  6. [6] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, I. Ann. Phys. (NY) 111 (1978), 61–110. Zbl0377.53024MR496157
  7. [7] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation quantization and quantization, II. Ann. Phys. (NY) 111 (1978), 111–151. Zbl0377.53025MR496158
  8. [8] Y. Belopolskaya and Y. K. Daletskii, “Stochastic Equations and Differential Geometry”, Kluwer, Dordrecht, 1990. Zbl0696.60053
  9. [9] Y. Belopolskaya and Y. Gliklikh, Stochastic processes on groups of diffeomorphisms and viscous hydrodynamics, Inf. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 145-159. Zbl1056.58010MR1914833
  10. [10] J. M. Bismut, “Mécanique Aléatoire”, Lect. Notes. Math., Vol. 966, Springer, Heidelberg, 1981. Zbl0457.60002
  11. [11] J. M. Bismut, “Large Deviations and Malliavin Calculus”, Progress in Math., Vol. 45, Birkhäuser, Basel, 1984. Zbl0537.35003MR755001
  12. [12] Z. Brzezniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds, Method. Funct. Anal. Topology 6 (in honour of Y. Daletskii) (2000), 43–84. Zbl0965.58028MR1784435
  13. [13] Z. Brzezniak and R. Léandre, Horizontal lift of an infinite dimensional diffusion, Potential Anal. 12 (2000), 249–280. Zbl0960.58020MR1752854
  14. [14] Z. Brzezniak and R. Léandre, Brownian pants on a manifold, Preprint. Zbl1103.58018
  15. [15] A. Cattaneo and G. Felder, A path integral approach to Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591–611. Zbl1038.53088MR1779159
  16. [16] Y. Daletskii, Measures and stochastic equations on infinite-dimensional manifolds, In: “Espaces de Lacets”, R. Léandre, S. Paycha and T. Wuerzbacher (eds.), Publi. Univ. Strasbourg, Strasbourg, 1996, 45–52. 
  17. [17] G. Dito and D. Sternheimer, Deformation quantization; genesis, developments and metamorphoses, In: “Deformation Quantization”, G. Halbout (ed.), IRMA Lectures Notes in Math. Phys., Vol. 1, Walter de Gruyter, Berlin, 2002, 9–54. Zbl1014.53054MR1914780
  18. [18] D. Elworthy and D. Truman, Classical mechanics, the diffusion heat equation and the Schrödinger equation, J. Math. Phys. 22 (1981), 2144–2166. Zbl0485.70024MR641455
  19. [19] M. Gradinaru, F. Russo and P. Vallois, Generalized covariations, local time and Stratonovitch-Itô’s formula for fractional Brownian motion, Ann. Probab. 31 (2003), 1772–1820. Zbl1059.60067MR2016600
  20. [20] A. Hirshfeld, Deformation quantization in quantum mechanics and quantum field theory, In: “Geometry, Integrability and Quantization. IV”, Mladenov I. and Naber G. (eds.), Coral Press, Sofia, 2003, 11–38. Zbl1039.53104MR1977559
  21. [21] A. Hirshfeld and T. Schwarzweller, Path integral quantization of the Poisson-Sigma model, Ann. Phys. (Leipzig), 9 (2000), 83–101. Zbl1017.81039MR1758657
  22. [22] J. Jacod, “Calcul Stochastique et Problemes de Martingales”, Lect. Notes. Math., Vol. 714, Springer, Heidelberg, 1975. Zbl0414.60053MR542115
  23. [23] J. R. Klauder and S. V. Shabanov, An introduction to coordinate-free quantization and its application to constrained systems, In: “Mathematical Methods of Quantum Physics”, C. C. Bernido (ed.), Gordon and Breach., Amsterdam, 1999, 117–131. Zbl1170.81398MR1723670
  24. [24] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., to appear. Zbl1058.53065MR2062626
  25. [25] H. Kunita, “Stochastic Flows and Stochastic Differential Equations”, Camb. Univ. Press, Cambridge, 1990. Zbl0743.60052MR1070361
  26. [26] H. H. Kuo, Diffusion and Brownian motion on infinite dimensional manifolds, Trans. Amer. Math. Soc. 159 (1972), 439–451. Zbl0255.60057MR309206
  27. [27] S. Kusuoka, More recent theory of Malliavin Calculus, Sūgaku 5 (1992), 155–173. Zbl0796.60059MR1207203
  28. [28] R. Léandre, Applications quantitatives et qualitatives du Calcul de Malliavin, In: “French-Japanese Seminar”, Métivier M. and Watanabe S. (eds.). Lect. Notes. Math., Vol. 1322, Springer, Berlin, 1988, 109–123. English translation in: “Geometry of Random Motion”, Durrett R. and Pinsky M. (eds.) Contem. Math. 73, A.M.S., Providence, 1988, 173–197. Zbl0671.58044
  29. [29] R. Léandre, Cover of the Brownian bridge and stochastic symplectic action, Rev. Math. Phys. 12 (2000), 91–137. Zbl0968.58027MR1750777
  30. [30] R. Léandre, Analysis on loop spaces and topology, Math. Notes. 72 (2002), 212–229. Zbl1042.58003MR1942549
  31. [31] R. Léandre, Stochastic Wess-Zumino-Novikov-Witten model on the torus, J. Math. Phys. 44 (2003), 5530–5568. Zbl1063.58022MR2023542
  32. [32] R. Léandre, Brownian cylinders and intersecting branes, Rep. Math. Phys. 52 (2003), 363–372. Zbl1049.58036MR2029767
  33. [33] R. Léandre, Markov property and operads, In: “Quantum Limits in the Second Law of Thermodynamics”, I. Nikulov and D. Sheehan (eds.), Entropy, Vol. 6, 2004, 180–215. Zbl1063.60076MR2081873
  34. [34] R. Léandre, Brownian pants and Deligne cohomology, J. Math. Phys. 46 (2005). Zbl1067.58029MR2125578
  35. [35] R. Léandre, Bundle gerbes and Brownian motion, In: “Lie Theory and Application in Physics. V.”, V. Dobrev and H. Doebner (eds.). World Scientific, Singapore, 2004, 343–352. MR2172912
  36. [36] R. Léandre, Galton-Watson tree and branching loop, In: “Geometry, Integrability and Quantization. VI”, I. Mladenov and A. Hirshfeld (eds.), Softek, Sofia, 2005, 276–284. Zbl1078.60067MR2161774
  37. [37] R. Léandre, Two examples of stochastic field theories, Osaka J. Math. 42 (2005), 353–365. Zbl1078.81056MR2147732
  38. [38] P. Malliavin, Stochastic Calculus of variation and hypoelliptic operators, In: “Stochastic Analysis”, K. Itô (ed.), Kinokuyina, Tokyo, 1978, 155-263. Zbl0411.60060MR536013
  39. [39] P. A. Meyer, Flot d’une équation différentielle stochastique, In: “Séminaire de Probabilités. XV”, Azéma J. and Yor M. (eds.), Lect. Notes. Math., Vol. 850, Springer, Heidelberg, 1981, 100–117. Zbl0461.60076MR622556
  40. [40] S. Molchanov, Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975), 1–63. Zbl0315.53026MR413289
  41. [41] D. Nualart, “Malliavin Calculus and Related Topics”, Springer, Heidelberg, 1997. Zbl0837.60050MR2200233
  42. [42] D. Nualart and M. Sanz, Malliavin Calculus for two-parameter Wiener functionals, Z. Wahrsch. Verw. Gebiete 70 (1985), 573–590. Zbl0595.60065MR807338
  43. [43] V. Pipiras and M. Taqqu, Integration question related to fractional Brownian motion, Probab. Theory Related Fields 118 (2000), 251–291. Zbl0970.60058MR1790083
  44. [44] H. K. Sugita, Positive generalized Wiener functions and potential theory over abstract Wiener spaces, Osaka J. Math. 25 (1988), 665–696. Zbl0737.46038MR969026
  45. [45] S. Watanabe, Stochastic analysis and its application Sūgaku 5 (1992), 51–71. Zbl0796.60058MR1161472

NotesEmbed ?

top

You must be logged in to post comments.