Existence, uniqueness and ergodicity for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions

In this paper, we prove the existence and uniqueness of solutions as well as ergodicity for McKean-Vlasov SDEs under Lyapunov conditions, in which the Lyapunov functions are defined on $\mathbb R^d\times \mathcal P_2(\mathbb R^d)$, i.e. the Lyapunov functions depend not only on space variable but also on distribution variable. It is reasonable and natural to consider distribution-dependent Lyapunov functions since the coefficients depends on distribution variable. We apply the martingale representation theorem and a modified Yamada-Watanabe theorem to obtain the existence and uniqueness of solutions. Furthermore, the Krylov-Bogolioubov theorem is used to get ergodicity since it is valid by linearity of the corresponding Fokker-Planck equations on $\mathbb R^d\times \mathcal P_2(\mathbb R^d)$. In particular, if the Lyapunov function depends only on space variable, we obtain exponential ergodicity for semigroup $P_t^*$ under Wasserstein quasi-distance. Finally, we give some examples to illustrate our theoretical results.