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.