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 Zd, d ≥ 2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure ν. We model the system with interacting Brownian diffusions X = (Xi(t))t≥0;i∈Zd moving on circles. We prove that for small times t and arbitrary initial Gibbs measures ν, or for long times and both high- or infinite-temperature initial measure and dynamics, the evolved measure νt stays Gibbsian. Furthermore, we show that for a low-temperature initial measure ν evolving under infinite-temperature dynamics there is a time interval (t0, t1) such that νt fails to be Gibbsian for d ≥ 2.