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