## Details

Consider the following mean-field equation on R^d:

d X_t = V(X_t, mu_t) dt + d B_t, where mu_t is the law of X_t, the drift V(x, mu) is smooth and confining, and (B_t) is a standard Brownian motion.

This McKean-Vlasov equation may admit multiple invariant probability measures. I will discuss the (local) stability of one of these equilibria.

Using Lions derivatives, a stability criterion is derived, analogous to the Jacobian stability criterion for ODEs. Under this spectral condition, the equilibrium is shown to be attractive for the Wasserstein metric W1. In addition, I will discuss a metastable behavior of the associated particle system, around a stable equilibrium of the mean-field equation.