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

We study the Gibbsian character of time-evolved planar rotor systems (that is, systems which have two-component, classical XY, spins) on , d>=2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure [nu]. We model the system with interacting Brownian diffusions moving on circles. We prove that for small times t and arbitrary initial Gibbs measures [nu], or for long times and both high- or infinite-temperature initial 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 there is a time interval (t0,t1) such that [nu]t fails to be Gibbsian for d>=2.