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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

  • 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.

    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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  • 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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