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 magnetization of the spins via a general potential. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. As in \cite{FedHoMa13}, which considers the Curie-Weiss model with Ising spins interacting via a quadratic potential and subjected to independent spins flips, we follow the program outlined in \cite{vEFedHoRe10}. We show that in the thermodynamic limit the system is non-Gibbs at time $t \in (0,\infty)$ if and only if there exists an $\alpha \in \mathbb{R}$ such that the large deviation rate function for the trajectory of the magnetization conditional on hitting the value $\alpha$ at time $t$ has multiple global minimizers. We further show that different minimizing trajectories are different at time $t=0$. We give conditions on the potential under which the system is Gibbs at time $t=0$, classify the possible scenarios of being Gibbs at time $t \in (0,\infty)$ in terms of the second difference quotient of the potential, and show that the system cannot become Gibbs once it has become non-Gibbs, i.e., there is a unique and explicitly computable crossover time $t_c \in [0,\infty]$ from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness ($t_c=0$), short-time conservation of Gibbsianness, large-time loss of Gibbsianness ($t_c\in (0,\infty)$), and preservation of Gibbsianness ($t_c=\infty$). Depending on the potential, the system can be Gibbs or non-Gibbs at the cross-over time time $t=t_c$.