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

TL;DR

This paper establishes the existence, uniqueness, and ergodicity of solutions to McKean-Vlasov SDEs using distribution-dependent Lyapunov functions, extending classical results to more general settings.

Contribution

It introduces Lyapunov functions that depend on both space and distribution variables, and applies advanced theorems to prove key properties of McKean-Vlasov SDEs.

Findings

Existence and uniqueness of solutions under distribution-dependent Lyapunov conditions

Ergodicity results including exponential ergodicity under certain conditions

Application of martingale representation and Yamada-Watanabe theorems

Abstract

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.