  We consider infinite-dimensional diffusions where the interaction between the
coordinates has a finite extent both in space and time. In particular, it is
not supposed to be smooth or Markov. The initial state of the system is Gibbs,
given by a strong summable interaction. If the strongness of this initial
interaction is lower than a suitable level, and if the dynamical interaction is
bounded from above in a right way, we prove that the law of the diffusion at
any time t is a Gibbs measure with absolutely summable interaction. The main
tool is a cluster expansion in space uniformly in time of the Girsanov factor
coming from the dynamics and exponential ergodicity of the free dynamics to an
equilibrium product measure.
