Quantitative Harris type theorems for diffusions and McKean-Vlasov processes

We consider ᵈ-valued diffusion processes of type \{align*} dX_t\\ =\\ b(X_t)dt\\, +\\, dB_t. \{align*} Assuming a geometric drift condition, we establish contractions the transitions kernels in Kantorovich (L¹ Wasserstein) distances with constants. Our results are in the spirit of Hairer and Mattingly's of Harris' Theorem. In particular, they do not rely on a small set. Instead we combine Lyapunov functions with reflection coupling and distance functions. We retrieve constants that are explicit in which can be computed with little effort from one-sided Lipschitz for the drift coefficient and the growth of a chosen Lyapunov. Consequences include exponential convergence in weighted total norms, gradient bounds, bounds for ergodic averages, and Kantorovich for nonlinear McKean-Vlasov diffusions in the case of sufficiently but not necessarily bounded nonlinearities. We also establish quantitative for sub-geometric ergodicity assuming a sub-geometric drift condition.