A bifurcation analysis of some McKean-Vlasov equations

Abstract

We study the long time behavior of a family of McKean-Vlasov stochastic differential equations. We give conditions ensuring the local stability of an invariant probability measure. Our criterion involves the location of the roots of an explicit holomorphic function associated to the dynamics. When all the roots lie on the left-half plane, local stability holds and we prove convergence in Wasserstein norms. We also provide the optimal rate of convergence. Our probabilistic proof makes use of Lions derivatives and relies on a new "integrated sensibility" formula.