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.