Loss and Recovery of Gibbsianness for XY models in external fields

We consider planar rotors (XY spins) in $\mathbb{Z}^d$, starting from an initial Gibbs measure and evolving with infinite-temperature stochastic (diffusive) dynamics. At intermediate times, if the system starts at low temperature, Gibbsianness can be lost. Due to the influence of the external initial field, Gibbsianness can be recovered after large finite times. We prove some results supporting this picture.