McKean-Vlasov SDEs with Bounded Measurable Interaction

Abstract

In this paper, McKean-Vlasov SDEs with bounded measurable interaction is investigated. The regularity estimate $$\|P_t^\ast\gamma^1-P_t^\ast\gamma^2\|_{var}\leq ct^{-\frac{1}{2}}\W_{1}(\gamma^1,\gamma^2),\ \ t\in(0,T]$$ for the nonlinear semigroup $P_t^\ast$ associated to McKean-Vlasov SDEs is derived. Two cases are considered respectively. The first case concentrates on the model where the interaction in the drift is merely assumed to be bounded measurable while the distribution dependent diffusion term is allowed to be Lipschitz continuous under $L^\eta$($\eta\in(0,1)$)-Wasserstein distance in the measure variable. In the second case, the diffusion is distribution free and the drift contain two parts: a bounded measurable interaction term plus a partially dissipative term. As an application of the regularity estimate, the exponential ergodicity in $\W_1$ is obtained in the second case.