Quantitative Harris type theorems for diffusions and McKean-Vlasov processes

TL;DR

This paper develops explicit quantitative Harris-type theorems for diffusion processes and McKean-Vlasov processes, providing contraction rates and convergence bounds under geometric and sub-geometric conditions without relying on small set conditions.

Contribution

It introduces a new approach combining Lyapunov functions with reflection coupling to obtain explicit contraction constants for diffusions and McKean-Vlasov processes.

Findings

Explicit contraction constants in Wasserstein distances.

Exponential convergence and gradient bounds.

Quantitative bounds for sub-geometric ergodicity.

Abstract

We consider $\mathbb{R}^d$-valued diffusion processes of type \begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.