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Infinite system of Brownian balls with interaction: the non-reversible case

Published online by Cambridge University Press:  01 March 2007

Myriam Fradon  and
Sylvie Rœlly
Myriam Fradon
Affiliation:
Laboratoire CNRS 8524, UFR de Mathématiques, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France; [email protected]
Sylvie Rœlly
Affiliation:
Institut für Mathematik, Universität Potsdam, Am Neuen Palais, 14415 Potsdam, Germany; [email protected] On leave of absence Centre de Mathématiques Appliquées, UMR CNRS 7641, École Polytechnique, 91128 Palaiseau Cedex, France.
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Abstract

We consider an infinite system of hard balls in $\xR^d$ undergoing Brownian motionsand submitted to a smooth pair potential.It is modelized by an infinite-dimensional stochastic differential equationwith an infinite-dimensional local time term.Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also showthat Gibbs measures are reversible measures.

Keywords

Stochastic differential equationlocal timehard core potentialGibbs measurereversible measure.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 55 - 79
Copyright
© EDP Sciences, SMAI, 2007

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References

Dobrushin, R.L., Gibbsian random fields. The general case. Functional Anal. Appl. 3 (1969) 22–28. CrossRef
Fradon, M. and Rœlly, S., Infinite dimensional diffusion processes with singular interaction. Bull. Sci. math. 124 (2000) 287–318. CrossRef
Fritz, J., Gradient Dynamics of Infinite Points Systems. Ann Probab. 15 (1987) 478–514. CrossRef
H.-O. Georgii, Canonical Gibbs measures. Lecture Notes in Mathematics 760, Springer-Verlag, Berlin (1979).
Lang, R., Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. Z. Wahrsch. Verw. Geb. 38 (1977) 55–72. CrossRef
Ruelle, D., Superstable Interactions in Classical Statistical Mechanics. Comm. Math. Phys. 18 (1970) 127–159. CrossRef
Saisho, Y. and Tanaka, H., Stochastic Differential Equations for Mutually Reflecting Brownian Balls. Osaka J. Math. 23 (1986) 725–740.
Tanemura, H., System, A of Infinitely Many Mutually Reflecting Brownian Balls. Probability Theory and Related Fields 104 (1996) 399–426. CrossRef