Short-Time Gibbsianness for Infinite-Dimensional Diffusions with Space-Time Interaction

Abstract

We consider a class of infinite-dimensional diffusions where the interaction between
the components has a finite extent both in space and time. We start the system from
a Gibbs measure with a finite-range uniformly bounded interaction. Under suitable conditions
on the drift, we prove that there exists t0 > 0 such that the distribution at time t ≤ t0 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.