Stationary states of interacting Brownian motions

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.