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

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 $t\le t_0$ 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.