Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes
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- by Andreas Eberle, Arnaud Guillin and Raphael Zimmer PDF
- Trans. Amer. Math. Soc. 371 (2019), 7135-7173 Request permission
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 the 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 subgeometric ergodicity assuming a subgeometric drift condition.References
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Additional Information
- Andreas Eberle
- Affiliation: Universität Bonn Institut für Angewandte Mathematik Endenicher Allee 60 53115 Bonn, Germany
- MR Author ID: 363836
- Email: eberle@uni-bonn.de
- Arnaud Guillin
- Affiliation: Laboratoire de Mathématiques CNRS - UMR 6620 Université Blaise Pascal Avenue des landais, 63177 Aubiere cedex, France
- MR Author ID: 661909
- Email: guillin@math.univ-bpclermont.fr
- Raphael Zimmer
- Affiliation: Universität Bonn Institut für Angewandte Mathematik Endenicher Allee 60 53115 Bonn, Germany
- MR Author ID: 1228825
- Email: raphael@infozimmer.de
- Received by editor(s): October 11, 2017
- Received by editor(s) in revised form: February 21, 2018
- Published electronically: September 28, 2018
- Additional Notes: The authors gratefully acknowledge financial support from DAAD and French government through the PROCOPE program, and from the German Science foundation through the Hausdorff Center for Mathematics.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7135-7173
- MSC (2010): Primary 60J60; Secondary 60H10
- DOI: https://doi.org/10.1090/tran/7576
- MathSciNet review: 3939573