Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time tc∈[0,∞]tc∈[0,∞] from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness (tc=0tc=0), short-time conservation and large-time loss of Gibbsianness (tc∈(0,∞)tc∈(0,∞)), and preservation of Gibbsianness (tc=∞tc=∞). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time t=tct=tc.