On gradient structures for Markov chains and the       passage to Wasserstein gradient flows

Karoline Disser; Matthias Liero

doi:10.3934/nhm.2015.10.233

Article Contents

2015, Volume 10, Issue 2: 233-253. Doi: 10.3934/nhm.2015.10.233

This issue Previous Article Next Article Conservation law models for traffic flow on a network of roads

On gradient structures for Markov chains and the passage to Wasserstein gradient flows

Karoline Disser^1, and
Matthias Liero^2,

1.
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin
2.
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin

Received: February 28, 2014

Revised: September 30, 2014

Published: May 2015

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Abstract

We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

Keywords:

Wasserstein gradient flow,
discrete gradient flow structures,
entropy/entropy-dissipation formulation,
evolutionary $\Gamma$-convergence,
Markov chains.

Mathematics Subject Classification: 35K10, 35K20, 37L05, 49M25, 60F99, 65M08, 60J27, 70G75, 82B35.

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