Loss without recovery of Gibbsianness during diffusion of continuous spins

We consider a specific continuous-spin Gibbs distribution μt=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For ‘high temperature’ initial measures we prove that the time-evoved measure μt is Gibbsian for all t. For ‘low temperature’ initial measures we prove that μt stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d ≥ 2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.