# Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains

Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. IST Austria.

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*Thesis*|

*Published*|

*English*

Supervisor

Department

Series Title

IST Austria Thesis

Abstract

This thesis is based on three main topics: In the first part, we study convergence of discrete gradient flow structures associated with regular finite-volume discretisations of Fokker-Planck equations. We show evolutionary I convergence of the discrete gradient flows to the L2-Wasserstein gradient flow corresponding to the solution of a Fokker-Planck
equation in arbitrary dimension d >= 1. Along the argument, we prove Mosco- and I-convergence results for discrete energy functionals, which are of independent interest for convergence of equivalent gradient flow structures in Hilbert spaces.
The second part investigates L2-Wasserstein flows on metric graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein distance, which is proved via a regularisation scheme for solutions of the continuity equation, adapted to the peculiar geometric structure of metric graphs. Based on those results, we show that the L2-Wasserstein space over a metric graph admits a gradient flow which may be identified as a solution of a Fokker-Planck equation.
In the third part, we focus again on the discrete gradient flows, already encountered in the first part. We propose a variational structure which extends the gradient flow structure to Markov chains violating the detailed-balance conditions. Using this structure, we characterise contraction estimates for the discrete heat flow in terms of convexity of
corresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals.

Publishing Year

Date Published

2020-03-31

Page

154

ISSN

IST-REx-ID

### Cite this

Forkert DL. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. 2020. doi:10.15479/AT:ISTA:7629

Forkert, D. L. (2020).

*Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains*. IST Austria. https://doi.org/10.15479/AT:ISTA:7629Forkert, Dominik L. “Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains.” IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:7629.

D. L. Forkert, “Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains,” IST Austria, 2020.

Forkert, Dominik L.

*Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains*. IST Austria, 2020, doi:10.15479/AT:ISTA:7629.**All files available under the following license(s):**

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