### 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.

Original language | English |
---|---|

Pages (from-to) | 653-674 |

Number of pages | 22 |

Journal | Markov Processes and Related Fields |

Volume | 20 |

Issue number | 4 |

Publication status | Published - 2014 |

### Keywords

- infinite-dimensional diffusion
- cluster expansion
- non-Markov drift
- Girsanov formula
- ultracontractivity
- planar rotors

## Fingerprint Dive into the research topics of 'Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction'. Together they form a unique fingerprint.

## Cite this

Rœlly, S., & Ruszel, W. M. (2014). Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction.

*Markov Processes and Related Fields*,*20*(4), 653-674.