Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-23T16:10:23.286Z Has data issue: false hasContentIssue false

Interacting Brownian particles and Gibbs fields on pathspaces

Published online by Cambridge University Press:  15 May 2003

David Dereudre*
Centre de Mathématiques Appliquées, UMR 7641, École Polytechnique, 91128 Palaiseau Cedex, France;
Get access


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.

Research Article
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Albeverio, S., Kondratiev, Yu.G. and Röckner, M., Analysis and geometry on configuration spaces: The Gibbsian case. J. Funct. Anal. 157 (1998) 242-291. CrossRef
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.
Cattiaux, P., Rœlly, S. and Zessin, H., Une approche gibbsienne des diffusions browniennes infini-dimensionnelles. Probab. Theory Related Fields 104-2 (1996) 223-248.
Dai Pra, P., Rœlly, S. and Zessin, H., Gibbs, A variational principle in space-time for infinite-dimensional diffusions. Probab. Theory Related Fields 122 (2002) 289-315. CrossRef
Dereudre, D., Une caractérisation de champs de Gibbs canoniques sur $\ensuremath{{\mathbb{R}^d}} $ et $\ensuremath{\mathcal{C}} ([0,1],\ensuremath{{\mathbb{R}^d}} )$ . C. R. Acad. Sci. Paris Sér. I 335 (2002) 177-182. CrossRef
D. Dereudre, Diffusions infini-dimensionnelles et champs de Gibbs sur l'espace des trajectoires continues ${\cal{C}}([0,1);\ensuremath{{\mathbb{R}^d}} )$ . Thèse soutenue à l'École Polytechnique (2002).
Deuschel, J.D., Infinite dimensionnal diffusion processes as Gibbs measures on $C[0,1]^{\ensuremath{{\mathbb{Z}^d}} }$ . Probab. Theory Related Fields 76 (1987) 325-340. CrossRef
Dobrushin, R.L. and Fritz, J., Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction. Comm. Math. Phys. 55 (1977) 275-292. CrossRef
H. Föllmer, Time reversal on Wiener space. Springer-Verlag, Lecture Notes in Math. 1158 (1986) 117-129.
Föllmer, H. and Wakolbinger, A., Time reversal of infinite-dimensional diffusions. Stochastic Process. Appl. 22 (1986) 59-77. CrossRef
Fradon, M., Roelly, S. and Tanemura, H., An infinite system of Brownian balls with infinite range interaction. Stochastic Process. Appl. 90-1 (2000) 43-66. CrossRef
Fritz, J., Gradient dynamics of infinite point systems. Ann. Probab. 15 (1987) 487-514. CrossRef
Fritz, J. and Dobrushin, R.L., Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Comm. Math. Phys. 57 (1977) 67-81. CrossRef
Fritz, J., Rœlly, S. and Zessin, H., Stationary states of interacting Brownian motions. Stud. Sci. Math. Hung. 34 (1998) 151-164.
Gaveau, B. and Trauber, P., L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46 (1996) 230-238. CrossRef
H.-O. Georgii, Canonical Gibbs measures. Springer, Lecture Notes in Math. 760 (1979).
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.
E. Glötzl, Gibbsian description of point processes, in Colloquia Mathematica Societatis Janos Bolyai, 24 keszthely. Hungary (1978) 69-84.
Glötzl, E., Lokale Energien und Potentiale für Punktprozesse. Math. Nach. 96 (1980) 195-206. CrossRef
J. Jacod, Calcul stochastique et problèmes de matingales. Springer, Lecture Notes in Math. 714 (1979).
Lang, R., Unendlich-dimensionale Wienerprozesse mit Wechselwirkung I. Z. Wahrsch. Verw. Gebiete 38 (1977) 55-72. CrossRef
Lang, R., Unendlich-dimensionale Wienerprozesse mit Wechselwirkung II. Z. Wahrsch. Verw. Gebiete 39 (1977) 277-299. CrossRef
K. Matthes, J. Kerstan and J. Mecke, Infinitely Divisible Point Process. J. Wiley (1978).
Millet, A., Nualart, D. and Sanz, M., Time Reversal for infinite-dimensional diffusions. Probab. Theory Related Fields 82 (1989) 315-347. CrossRef
Minlos, R.A., Rœlly, S. and Zessin, H., Gibbs states on space-time. Potential Anal. 13 (2000) 367-408. CrossRef
Nguyen, X.X. and Zessin, H., Integral and differential characterizations of the Gibbs process. Math. Nach. 88 (1979) 105-115.
C. Preston, Random fields. Springer, Lecture Notes in Math. 714 (1976).
Privault, N., A characterization of grand canonical Gibbs measures by duality. Potential Anal. 15 (2001) 23-28. CrossRef
B. Rauchenschwandtner and A. Wakolbinger, Some aspects of the Papangelou kernel, in Colloquia mathematica societatis Janos Bolyai, 24 keszthely. Hungary (1978) 325-336.
Rœlly, S. and Zessin, H., Une caractérisation de champs gibbsiens sur un espace de trajectoires. C. R. Acad. Sci. Paris Sér. I 321 (1995) 1377-1382.
D. Ruelle, Statistical Mechanics. Rigorous Results.. Benjamin, New York (1969) .
Ruelle, D., Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 (1970) 127-159. CrossRef
Yoshida, M., Construction of infinite dimensional interacting diffusion processes through Dirichlet forms. Probab. Theory Related Fields 106 (1996) 265-297. CrossRef