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