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 Z(d), 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 X = (X(i)(t))(t >= 0,i is an element of Z)(d) 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 (t(0), t(1)) such that nu(t) fails to be Gibbsian for d >= 2. (C) 2008 Elsevier B.V. All rights reserved.