Stationary states of interacting Brownian motions

Citation
J. Fritz et al., Stationary states of interacting Brownian motions, ST SCI M H, 34(1-3), 1998, pp. 151-164
Citations number
20
Categorie Soggetti
Mathematics
Journal title
STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
ISSN journal
00816906 → ACNP
Volume
34
Issue
1-3
Year of publication
1998
Pages
151 - 164
Database
ISI
SICI code
0081-6906(1998)34:1-3<151:SSOIBM>2.0.ZU;2-W
Abstract
We are interested in a description of stationary states of gradient dynamic s of interacting Brownian particles. In contrast to lattice models, this pr oblem does not seem to be solvable at a formal level of the stationary Kolm agorov equation. We can only study stationary states of a well controlled M arkov process. In space dimensions four or less, for smooth and superstable pair potentials of finite range the non-equilibrium dynamics of interactin g Brownian particles can be constructed in an explicitly defined determinis tic set of locally finite configurations, see [Fr2]. This set is of full me asure with respect to any canonical Gibbs state for the interaction,and eve ry canonical state is a stationary one. Assuming translation invariance of a stationary measure, and also the finiteness of its specific entropy with respect to an equilibrium Gibbs state, it is shown that this stationary sta te is canonical Gibbs. Related ideas of Alfred Renyi and some of their cons equences are also reviewed.