On the stability of the invariant probability measures of McKean-Vlasov equations

We study the long-time behavior of some McKean-Vlasov stochastic differential equations used to model the evolution of large populations of interacting agents. We give conditions ensuring the local stability of an invariant probability measure. Lions derivatives are used in a novel way to obtain our stability criteria. We obtain results for non-local McKean-Vlasov equations on $\mathbb{R}^d$ and for McKean-Vlasov equations on the torus where the interaction kernel is given by a convolution. On $\mathbb{R}^d$, we prove that the location of the roots of an analytic function determines the stability. On the torus, our stability criterion involves the Fourier coefficients of the interaction kernel. In both cases, we prove the convergence in the Wasserstein metric $W_1$ with an exponential rate of convergence.