Interacting Brownian particles and Gibbs fields on pathspaces

David Dereudre

ESAIM: Probability and Statistics (2010)

  • Volume: 7, page 251-277
  • ISSN: 1292-8100

Abstract

top
In this paper, we prove that the laws of interacting Brownian particles are characterized as Gibbs fields on pathspace associated to an explicit class of Hamiltonian functionals. More generally, we show that a large class of Gibbs fields on pathspace corresponds to Brownian diffusions. Some applications to time reversal in the stationary and non stationary case are presented.

How to cite

top

Dereudre, David. "Interacting Brownian particles and Gibbs fields on pathspaces." ESAIM: Probability and Statistics 7 (2010): 251-277. <http://eudml.org/doc/104308>.

@article{Dereudre2010,
abstract = { In this paper, we prove that the laws of interacting Brownian particles are characterized as Gibbs fields on pathspace associated to an explicit class of Hamiltonian functionals. More generally, we show that a large class of Gibbs fields on pathspace corresponds to Brownian diffusions. Some applications to time reversal in the stationary and non stationary case are presented. },
author = {Dereudre, David},
journal = {ESAIM: Probability and Statistics},
keywords = {Point measure on pathspace; Gibbs field; interacting Brownian particles; integration by parts formula; Campbell measure.; point measure on pathspace; interacting Brownian particles; Campbell measure},
language = {eng},
month = {3},
pages = {251-277},
publisher = {EDP Sciences},
title = {Interacting Brownian particles and Gibbs fields on pathspaces},
url = {http://eudml.org/doc/104308},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Dereudre, David
TI - Interacting Brownian particles and Gibbs fields on pathspaces
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 251
EP - 277
AB - In this paper, we prove that the laws of interacting Brownian particles are characterized as Gibbs fields on pathspace associated to an explicit class of Hamiltonian functionals. More generally, we show that a large class of Gibbs fields on pathspace corresponds to Brownian diffusions. Some applications to time reversal in the stationary and non stationary case are presented.
LA - eng
KW - Point measure on pathspace; Gibbs field; interacting Brownian particles; integration by parts formula; Campbell measure.; point measure on pathspace; interacting Brownian particles; Campbell measure
UR - http://eudml.org/doc/104308
ER -

References

top
  1. S. Albeverio, Yu.G. Kondratiev and M. Röckner, Analysis and geometry on configuration spaces: The Gibbsian case. J. Funct. Anal.157 (1998) 242-291.  Zbl0931.58019
  2. S. Albeverio, M. Röckner and T.S. Zhang, Markov uniqueness for a class of infinite dimensional Dirichlet operators. Stochastic Process. Optimal Control, Stochastics Monogr. 7 (1993) 1-26.  Zbl0827.31007
  3. P. Cattiaux, S. Rœlly and H. Zessin, Une approche gibbsienne des diffusions browniennes infini-dimensionnelles. Probab. Theory Related Fields104-2 (1996) 223-248.  
  4. P. Dai Pra, S. Rœlly and H. Zessin, A Gibbs variational principle in space-time for infinite-dimensional diffusions. Probab. Theory Related Fields122 (2002) 289-315.  Zbl0998.60092
  5. D. Dereudre, Une caractérisation de champs de Gibbs canoniques sur d et 𝒞 ( [ 0 , 1 ] , d ) . C. R. Acad. Sci. Paris Sér. I335 (2002) 177-182.  
  6. D. Dereudre, Diffusions infini-dimensionnelles et champs de Gibbs sur l'espace des trajectoires continues 𝒞 ( [ 0 , 1 ) ; d ) . Thèse soutenue à l'École Polytechnique (2002).  
  7. J.D. Deuschel, Infinite dimensionnal diffusion processes as Gibbs measures on C [ 0 , 1 ] d . Probab. Theory Related Fields76 (1987) 325-340.  
  8. R.L. Dobrushin and J. Fritz, Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction. Comm. Math. Phys.55 (1977) 275-292.  Zbl0987.82502
  9. H. Föllmer, Time reversal on Wiener space. Springer-Verlag, Lecture Notes in Math. 1158 (1986) 117-129.  Zbl0582.60078
  10. H. Föllmer and A. Wakolbinger, Time reversal of infinite-dimensional diffusions. Stochastic Process. Appl.22 (1986) 59-77.  Zbl0602.60067
  11. M. Fradon, S. Roelly and H. Tanemura, An infinite system of Brownian balls with infinite range interaction. Stochastic Process. Appl.90-1 (2000) 43-66.  Zbl1046.60055
  12. J. Fritz, Gradient dynamics of infinite point systems. Ann. Probab.15 (1987) 487-514.  Zbl0623.60119
  13. J. Fritz and R.L. Dobrushin, Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Comm. Math. Phys.57 (1977) 67-81.  
  14. J. Fritz, S. Rœlly and H. Zessin, Stationary states of interacting Brownian motions. Stud. Sci. Math. Hung.34 (1998) 151-164.  Zbl0926.60080
  15. B. Gaveau and P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal.46 (1996) 230-238.  Zbl0488.60068
  16. H.-O. Georgii, Canonical Gibbs measures. Springer, Lecture Notes in Math. 760 (1979).  
  17. H.-O. Georgii, Equilibria for particle motions: Conditionally balanced point random fields, Exchangeability in Probability and Statistics, edited by Koch, Spizzichino. North Holland (1982) 265-280.  Zbl0495.60054
  18. E. Glötzl, Gibbsian description of point processes, in Colloquia Mathematica Societatis Janos Bolyai, 24 keszthely. Hungary (1978) 69-84.  
  19. E. Glötzl, Lokale Energien und Potentiale für Punktprozesse. Math. Nach.96 (1980) 195-206.  Zbl0455.60084
  20. J. Jacod, Calcul stochastique et problèmes de matingales. Springer, Lecture Notes in Math. 714 (1979).  Zbl0414.60053
  21. R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung I. Z. Wahrsch. Verw. Gebiete38 (1977) 55-72.  Zbl0349.60103
  22. R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung II. Z. Wahrsch. Verw. Gebiete39 (1977) 277-299.  Zbl0342.60067
  23. K. Matthes, J. Kerstan and J. Mecke, Infinitely Divisible Point Process. J. Wiley (1978).  Zbl0383.60001
  24. A. Millet, D. Nualart and M. Sanz, Time Reversal for infinite-dimensional diffusions. Probab. Theory Related Fields82 (1989) 315-347.  Zbl0659.60107
  25. R.A. Minlos, S. Rœlly and H. Zessin, Gibbs states on space-time. Potential Anal.13 (2000) 367-408.  Zbl0969.60065
  26. X.X. Nguyen and H. Zessin, Integral and differential characterizations of the Gibbs process. Math. Nach.88 (1979) 105-115.  Zbl0444.60040
  27. C. Preston, Random fields. Springer, Lecture Notes in Math. 714 (1976).  Zbl0335.60074
  28. N. Privault, A characterization of grand canonical Gibbs measures by duality. Potential Anal.15 (2001) 23-28.  Zbl0994.60057
  29. B. Rauchenschwandtner and A. Wakolbinger, Some aspects of the Papangelou kernel, in Colloquia mathematica societatis Janos Bolyai, 24 keszthely. Hungary (1978) 325-336.  
  30. S. Rœlly and H. Zessin, Une caractérisation de champs gibbsiens sur un espace de trajectoires. C. R. Acad. Sci. Paris Sér. I321 (1995) 1377-1382.  
  31. D. Ruelle, Statistical Mechanics. Rigorous Results.. Benjamin, New York (1969) .  Zbl0177.57301
  32. D. Ruelle, Superstable interactions in classical statistical mechanics. Comm. Math. Phys.18 (1970) 127-159.  Zbl0198.31101
  33. M. Yoshida, Construction of infinite dimensional interacting diffusion processes through Dirichlet forms. Probab. Theory Related Fields106 (1996) 265-297.  Zbl0859.60068

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.