Quentin Cormier, Inria Saclay Centre
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 \(W_1\). In addition, I will discuss a metastable behavior of the associated particle system, around a stable equilibrium of the mean-field equation.