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 X Y , spins) on Z d , 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 = ( X i ( t ) ) t ≥ 0 , i ∈ Z d 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 ν t stays Gibbsian. Furthermore, we show that for a low-temperature initial measure ν evolving under infinite-temperature dynamics there is a time interval ( t 0 , t 1 ) such that ν t fails to be Gibbsian for d ≥ 2 .