Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction

TL;DR

This paper proves that certain infinite-dimensional diffusions with space-time interactions preserve Gibbsian structure shortly after initial time, using cluster expansion techniques to analyze the dynamics.

Contribution

It introduces conditions under which the distribution of infinite-dimensional diffusions remains Gibbsian for small times, extending understanding of Gibbs measures under stochastic dynamics.

Findings

Existence of a small time interval where the distribution remains Gibbsian.

Use of cluster expansion to analyze space-time interactions.

Conditions on drift ensuring Gibbsian preservation.

Abstract

We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finite-range uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists $t_0>0$ such that the distribution at time $t\leq t_0$ is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics.