Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes

  • Eberle A
  • Guillin A
  • Zimmer R
87Citations
Citations of this article
19Readers
Mendeley users who have this article in their library.

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.

Cite

CITATION STYLE

APA

Eberle, A., Guillin, A., & Zimmer, R. (2018). Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes. Transactions of the American Mathematical Society, 371(10), 7135–7173. https://doi.org/10.1090/tran/7576

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free