Propagation of Gibbsianness for Infinite-Dimensional Diffusions with Space-Time Interaction
2014, v.20, Issue 4, 653-674
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.
Keywords: infinite-dimensional diffusion,cluster expansion,non-Markov drift,Girsanov formula,ultracontractivity,planar rotors