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

Karoline Disser; Matthias Liero; Karoline Disser; Matthias Liero

doi:10.3934/nhm.2015.10.233

Networks and Heterogeneous Media

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

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

Karoline Disser ^{1
,},
Matthias Liero ^{2
,}

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

Received: 01 March 2014 Revised: 01 October 2014
35K10, 35K20, 37L05, 49M25, 60F99, 65M08, 60J27, 70G75, 82B35.

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.
- Wasserstein gradient flow,
- discrete gradient flow structures,
- entropy/entropy-dissipation formulation,
- evolutionary $\Gamma$-convergence,
- Markov chains
Citation: Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows[J]. Networks and Heterogeneous Media, 2015, 10(2): 233-253. doi: 10.3934/nhm.2015.10.233

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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.

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