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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Andreas Eberle, Arnaud Guillin and Raphael Zimmer
Journal: Trans. Amer. Math. Soc. 371 (2019), 7135-7173
MSC (2010): Primary 60J60; Secondary 60H10
DOI: https://doi.org/10.1090/tran/7576
Published electronically: September 28, 2018
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Abstract: We consider $ {\mathbb{R}}^d$-valued diffusion processes of type

$\displaystyle dX_t\ =\ b(X_t)dt\, +\, dB_t.$    

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.

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Additional Information

Andreas Eberle
Affiliation: Universität Bonn Institut für Angewandte Mathematik Endenicher Allee 60 53115 Bonn, Germany
Email: eberle@uni-bonn.de

Arnaud Guillin
Affiliation: Laboratoire de Mathématiques CNRS - UMR 6620 Université Blaise Pascal Avenue des landais, 63177 Aubiere cedex, France
Email: guillin@math.univ-bpclermont.fr

Raphael Zimmer
Affiliation: Universität Bonn Institut für Angewandte Mathematik Endenicher Allee 60 53115 Bonn, Germany
Email: raphael@infozimmer.de

DOI: https://doi.org/10.1090/tran/7576
Keywords: Couplings, Wasserstein distances, Lyapunov functions, Harris theorem, quantitative bounds, convergence to stationarity, nonlinear diffusions
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.
Article copyright: © Copyright 2018 American Mathematical Society