Log-Sobolev Inequality for Decoupled and McKean-Vlasov SDEs and Application on Exponential Ergodicity

Abstract: The exponential ergodicity in the ( L^1 )-Wasserstein distance for partially dissipative McKean-Vlasov SDEs has been extensively studied. However, the question of exponential ergodicity in the ( L^2 )-Wasserstein distance and relative entropy has remained unresolved. This paper addresses the problem by establishing the log-Sobolev inequality for both the time-marginal distributions and the invariant probability measure, providing a positive resolution. As part of the groundwork, the log-Sobolev inequality is investigated for the associated time-inhomogeneous semigroup. The main results are further extended to degenerate diffusion.