WIAS Preprint No. 1899, (2013)

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



Authors

  • Disser, Karoline
  • Liero, Matthias
    ORCID: 0000-0002-0963-2915

2010 Mathematics Subject Classification

  • 35K10 35K20 37L05 49M25 60F99 65M08 60J27 70G75

Keywords

  • Wasserstein gradient flow, relative entropy, finite-volume scheme, entropy/entropy-dissipation formulation, gradient structures, Markov chains

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.

Appeared in

  • Netw. Heterog. Media, 10 (2015) pp. 233-253.

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