--- _id: '7629' abstract: - lang: eng text: "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\r\nequation 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.\r\nThe 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.\r\nIn 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\r\ncorresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Dominik L full_name: Forkert, Dominik L id: 35C79D68-F248-11E8-B48F-1D18A9856A87 last_name: Forkert citation: ama: 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 apa: Forkert, D. L. (2020). Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7629 chicago: Forkert, Dominik L. “Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7629. ieee: D. L. Forkert, “Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains,” Institute of Science and Technology Austria, 2020. ista: Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. Institute of Science and Technology Austria. mla: Forkert, Dominik L. Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7629. short: D.L. Forkert, Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains, Institute of Science and Technology Austria, 2020. date_created: 2020-04-02T06:40:23Z date_published: 2020-03-31T00:00:00Z date_updated: 2023-09-07T13:03:12Z day: '31' ddc: - '510' degree_awarded: PhD department: - _id: JaMa doi: 10.15479/AT:ISTA:7629 ec_funded: 1 file: - access_level: open_access checksum: c814a1a6195269ca6fe48b0dca45ae8a content_type: application/pdf creator: dernst date_created: 2020-04-14T10:47:59Z date_updated: 2020-07-14T12:48:01Z file_id: '7657' file_name: Thesis_Forkert_PDFA.pdf file_size: 3297129 relation: main_file - access_level: closed checksum: ceafb53f923d1b5bdf14b2b0f22e4a81 content_type: application/x-zip-compressed creator: dernst date_created: 2020-04-14T10:47:59Z date_updated: 2020-07-14T12:48:01Z file_id: '7658' file_name: Thesis_Forkert_source.zip file_size: 1063908 relation: source_file file_date_updated: 2020-07-14T12:48:01Z has_accepted_license: '1' language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: '154' project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria status: public supervisor: - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 title: Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2020' ...