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.
@article{ASNSP_2005_5_4_4_653_0,
author = {L\'eandre, R\'emi},
title = {Stochastic {Poisson-Sigma} model},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {653--667},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 4},
number = {4},
year = {2005},
mrnumber = {2207738},
zbl = {1170.53317},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2005_5_4_4_653_0/}
}
TY - JOUR AU - Léandre, Rémi TI - Stochastic Poisson-Sigma model JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 653 EP - 667 VL - 4 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2005_5_4_4_653_0/ LA - en ID - ASNSP_2005_5_4_4_653_0 ER -
Léandre, Rémi. Stochastic Poisson-Sigma model. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 653-667. http://archive.numdam.org/item/ASNSP_2005_5_4_4_653_0/
[1] and , “Quasi-sure Analysis”, Publication Université Paris VI, Paris, 1990.
[2] and , “Integration on Loop Groups”, Publication Université Paris VI, Paris, 1990.
[3] , 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] , and , Construction of a rotational invariant diffusion on the free loop space. C.R. Acad. Sci. Paris Sér. I Math. 316 (1993), 287-292. | MR | Zbl
[5] and , Stochastic tools on Hilbert manifolds: interplay with geometry and physics, Comm. Math. Phys. 197 (1997), 243-260. | MR | Zbl
[6] , , , and , Deformation theory and quantization, I. Ann. Phys. (NY) 111 (1978), 61-110. | MR | Zbl
[7] , , , and , Deformation quantization and quantization, II. Ann. Phys. (NY) 111 (1978), 111-151. | MR | Zbl
[8] and , “Stochastic Equations and Differential Geometry”, Kluwer, Dordrecht, 1990. | Zbl
[9] and , Stochastic processes on groups of diffeomorphisms and viscous hydrodynamics, Inf. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 145-159. | MR | Zbl
[10] , “Mécanique Aléatoire”, Lect. Notes. Math., Vol. 966, Springer, Heidelberg, 1981. | Zbl
[11] , “Large Deviations and Malliavin Calculus”, Progress in Math., Vol. 45, Birkhäuser, Basel, 1984. | MR | Zbl
[12] and , Stochastic differential equations on Banach manifolds, Method. Funct. Anal. Topology 6 (in honour of Y. Daletskii) (2000), 43-84. | MR | Zbl
[13] and , Horizontal lift of an infinite dimensional diffusion, Potential Anal. 12 (2000), 249-280. | MR | Zbl
[14] and , Brownian pants on a manifold, Preprint.
[15] and , A path integral approach to Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611. | MR | Zbl
[16] , 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] and , 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. | MR | Zbl
[18] and , Classical mechanics, the diffusion heat equation and the Schrödinger equation, J. Math. Phys. 22 (1981), 2144-2166. | MR | Zbl
[19] , and , Generalized covariations, local time and Stratonovitch-Itô's formula for fractional Brownian motion, Ann. Probab. 31 (2003), 1772-1820. | MR | Zbl
[20] , 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. | MR | Zbl
[21] and , Path integral quantization of the Poisson-Sigma model, Ann. Phys. (Leipzig), 9 (2000), 83-101. | MR | Zbl
[22] , “Calcul Stochastique et Problemes de Martingales”, Lect. Notes. Math., Vol. 714, Springer, Heidelberg, 1975. | MR | Zbl
[23] and , 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. | MR | Zbl
[24] , Deformation quantization of Poisson manifolds, Lett. Math. Phys., to appear. | MR | Zbl
[25] , “Stochastic Flows and Stochastic Differential Equations”, Camb. Univ. Press, Cambridge, 1990. | MR | Zbl
[26] , Diffusion and Brownian motion on infinite dimensional manifolds, Trans. Amer. Math. Soc. 159 (1972), 439-451. | MR | Zbl
[27] , More recent theory of Malliavin Calculus, Sūgaku 5 (1992), 155-173. | MR | Zbl
[28] , 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. | Zbl
[29] , Cover of the Brownian bridge and stochastic symplectic action, Rev. Math. Phys. 12 (2000), 91-137. | MR | Zbl
[30] , Analysis on loop spaces and topology, Math. Notes. 72 (2002), 212-229. | MR | Zbl
[31] , Stochastic Wess-Zumino-Novikov-Witten model on the torus, J. Math. Phys. 44 (2003), 5530-5568. | MR | Zbl
[32] , Brownian cylinders and intersecting branes, Rep. Math. Phys. 52 (2003), 363-372. | MR | Zbl
[33] , 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. | MR | Zbl
[34] , Brownian pants and Deligne cohomology, J. Math. Phys. 46 (2005). | MR | Zbl
[35] , 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. | MR
[36] , Galton-Watson tree and branching loop, In: “Geometry, Integrability and Quantization. VI”, I. Mladenov and A. Hirshfeld (eds.), Softek, Sofia, 2005, 276-284. | MR | Zbl
[37] , Two examples of stochastic field theories, Osaka J. Math. 42 (2005), 353-365. | MR | Zbl
[38] , Stochastic Calculus of variation and hypoelliptic operators, In: “Stochastic Analysis”, K. Itô (ed.), Kinokuyina, Tokyo, 1978, 155-263. | MR | Zbl
[39] , 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. | Numdam | MR | Zbl
[40] , Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975), 1-63. | MR | Zbl
[41] , “Malliavin Calculus and Related Topics”, Springer, Heidelberg, 1997. | MR | Zbl
[42] and , Malliavin Calculus for two-parameter Wiener functionals, Z. Wahrsch. Verw. Gebiete 70 (1985), 573-590. | MR | Zbl
[43] and , Integration question related to fractional Brownian motion, Probab. Theory Related Fields 118 (2000), 251-291. | MR | Zbl
[44] , Positive generalized Wiener functions and potential theory over abstract Wiener spaces, Osaka J. Math. 25 (1988), 665-696. | MR | Zbl
[45] , Stochastic analysis and its application Sūgaku 5 (1992), 51-71. | MR | Zbl
