Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction

Journal article (2014)

Authors

S. Rœlly University of Potsdam

W.M. Ruszel Applied Probability -

Research Group

Applied Probability () (TU Delft)

Cluster expansion Infinite-dimensional diffusion Non-Markov drift Girsanov formula Ultracontractivity Planar rotors

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Published Date

2014

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

Abstract

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.