Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

We consider Ising-spin systems starting from an initial Gibbs measure 1) and evolving under a spin-flip dynamics towards a reversible Gibbs measure mu not equal nu. Both v and mu are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure nuS(t) at time t and show the following:(1) For all nu and mu, nuS (t) is Gibbs for small t. (2) If both nu and mu have a high or infinite temperature, then nuS(t) is Gibbs for all t > 0. (3) If nu has a low non-zero temperature and a zero magnetic field and mu has a high or infinite temperature, then nuS(t) is Gibbs for small t and non-Gibbs for large t. (4) If nu has a low non-zero temperature and a non-zero magnetic field and mu has a high or infinite temperature, then nuS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where mu has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.