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Une approche Gibbsienne des diffusions Browniennes infini-dimensionnelles
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  • Published: June 1996

Une approche Gibbsienne des diffusions Browniennes infini-dimensionnelles

  • P. Cattiaux1,2,
  • S. Roelly3 &
  • H. Zessin4 

Probability Theory and Related Fields volume 104, pages 147–179 (1996)Cite this article

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  • 16 Citations

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Summary

We establish a one-to-one correspondence between the laws of smooth infinite dimensional brownian diffusions and the Gibbs states on the path space\(\Omega = C(0,1)^{\mathbb{Z}^d }\). Applications to phase transition, time reversal and reversible measures are then discussed. The main tool is a characterization of Gibbs states via an infinite dimensional version of the Malliavin calculus integration by parts formula.

Résumé

Nous montrons qu'il y a correspondance biunivoque entre les lois de diffusions browniennes infini-dimensionnelles et les états de Gibbs sur l'espace des trajectoires\(\Omega = C(0,1)^{\mathbb{Z}^d }\). Ce résultat est ensuite appliqué aux problèmes de transition de phases, du retournement du temps et à l'étude des mesures réversibles. Le principal outil est une caractérisation des états de Gibbs par une version infini-dimensionnelle de la formule d'intégration par parties du calcul de Malliavin.

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Authors and Affiliations

  1. C.M.A.P, U.R.A. C.N.R.S. 756, Ecole Polytechnique, F-91128, Palaiseau Cedex, France

    P. Cattiaux

  2. Equipe MODAL'X, U.F.R. SEGMI, Université Paris X Nanterre, 200 av. de la République, F-92001, Nanterre Cedex, France

    P. Cattiaux

  3. Laboratoire de Géométrie, Analyse et Topologie, U.R.A. C.N.R.S. 751, U.F.R. de Mathématiques, Université Lille 1, F-59655, Villeneuve d'Ascq Cedex, France

    S. Roelly

  4. Laboratoire de Statistiques et Probabilité, U.F.R. de Mathématiques, Université Lille 1, F-59655, Villeneuve d'Ascq Cedex, France

    H. Zessin

Authors
  1. P. Cattiaux
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  2. S. Roelly
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  3. H. Zessin
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Cattiaux, P., Roelly, S. & Zessin, H. Une approche Gibbsienne des diffusions Browniennes infini-dimensionnelles. Probab. Th. Rel. Fields 104, 147–179 (1996). https://doi.org/10.1007/BF01247836

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  • Received: 15 March 1993

  • Revised: 05 November 1994

  • Issue Date: June 1996

  • DOI: https://doi.org/10.1007/BF01247836

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Mathematics Subject Classification

  • 60H07
  • 60J60
  • 60K35
  • 82C22
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