Gibbsianness versus Non-Gibbsianness of time-evolved planar rotor models

We study the Gibbsian character of time-evolved planar rotor systems on Z^d, d at least 2, in the transient regime, evolving with stochastic dynamics and starting with an initial Gibbs measure. We model the system by interacting Brownian diffusions, moving on circles. We prove that for small times and arbitrary initial Gibbs measures \nu, or for long times and both high- or infinite-temperature measure and dynamics, the evolved measure \nu^t stays Gibbsian. Furthermore we show that for a low-temperature initial measure \nu, evolving under infinite-temperature dynamics thee is a time interval (t_0, t_1) such that \nu^t fails to be Gibbsian in d=2.