Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus

Helge Dietert; Josephine Evans; Thomas Holding

doi:10.3934/krm.2018056

Article Contents

2018, Volume 11, Issue 6: 1427-1441. Doi: 10.3934/krm.2018056

This issue Previous Article Numerical study of an anisotropic Vlasov equation arising in plasma physics Next Article Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Received: March 31, 2016

Revised: June 30, 2017

Published: June 2018

The authors were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis).

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Abstract

We study contraction for the kinetic Fokker-Planck operator on the torus. Solving the stochastic differential equation, we show contraction and therefore exponential convergence in the Monge-Kantorovich-Wasserstein $ \mathcal{W}_2$ distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.

Keywords:

Convergence to equilibrium,
hypocoercivity,
Monge-Kantorovich-Wasserstein $ \mathcal{W}_2$ distance,
Fokker-Planck equation,
torus,
co-adapted couplings.

Mathematics Subject Classification: 60J60; 35Q84, 60H30, 82C31.

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References

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