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

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


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