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

Myriam Fradon; Sylvie Rœlly

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 55-79
  • ISSN: 1292-8100

Abstract

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We consider an infinite system of hard balls in d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with 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 show that Gibbs measures are reversible measures.

How to cite

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Fradon, Myriam, and Rœlly, Sylvie. "Infinite system of Brownian balls with interaction: the non-reversible case." ESAIM: Probability and Statistics 11 (2007): 55-79. <http://eudml.org/doc/250111>.

@article{Fradon2007,
abstract = { We consider an infinite system of hard balls in $\xR^d$ undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with 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 show that Gibbs measures are reversible measures. },
author = {Fradon, Myriam, Rœlly, Sylvie},
journal = {ESAIM: Probability and Statistics},
keywords = {Stochastic differential equation; local time; hard core potential; Gibbs measure; reversible measure.; stochastic differential equation; Gibbs measure; reversible measure},
language = {eng},
month = {3},
pages = {55-79},
publisher = {EDP Sciences},
title = {Infinite system of Brownian balls with interaction: the non-reversible case},
url = {http://eudml.org/doc/250111},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Fradon, Myriam
AU - Rœlly, Sylvie
TI - Infinite system of Brownian balls with interaction: the non-reversible case
JO - ESAIM: Probability and Statistics
DA - 2007/3//
PB - EDP Sciences
VL - 11
SP - 55
EP - 79
AB - We consider an infinite system of hard balls in $\xR^d$ undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with 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 show that Gibbs measures are reversible measures.
LA - eng
KW - Stochastic differential equation; local time; hard core potential; Gibbs measure; reversible measure.; stochastic differential equation; Gibbs measure; reversible measure
UR - http://eudml.org/doc/250111
ER -

References

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  1. R.L. Dobrushin, Gibbsian random fields. The general case. Functional Anal. Appl.3 (1969) 22–28.  Zbl0192.61801
  2. M. Fradon and S. Rœlly, Infinite dimensional diffusion processes with singular interaction. Bull. Sci. math. 124 (2000) 287–318.  Zbl0958.60059
  3. J. Fritz, Gradient Dynamics of Infinite Points Systems. Ann Probab.15 (1987) 478–514.  Zbl0623.60119
  4. H.-O. Georgii, Canonical Gibbs measures. Lecture Notes in Mathematics 760, Springer-Verlag, Berlin (1979).  
  5. R. Lang, Unendlich-dimensionale Wienerprozesse mit Wechselwirkung. Z. Wahrsch. Verw. Geb.38 (1977) 55–72.  Zbl0349.60103
  6. D. Ruelle, Superstable Interactions in Classical Statistical Mechanics. Comm. Math. Phys.18 (1970) 127–159.  Zbl0198.31101
  7. Y. Saisho and H. Tanaka, Stochastic Differential Equations for Mutually Reflecting Brownian Balls. Osaka J. Math. 23 (1986) 725–740.  Zbl0613.60057
  8. H. Tanemura, A System of Infinitely Many Mutually Reflecting Brownian Balls. Probability Theory and Related Fields104 (1996) 399–426.  Zbl0849.60087

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