[{"volume":35,"intvolume":" 35","author":[{"full_name":"Khudiakova, Kseniia","id":"4E6DC800-AE37-11E9-AC72-31CAE5697425","last_name":"Khudiakova","orcid":"0000-0002-6246-1465","first_name":"Kseniia"},{"full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"last_name":"Pedrotti","full_name":"Pedrotti, Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","first_name":"Francesco"}],"isi":1,"issue":"3","_id":"20050","external_id":{"arxiv":["2402.04151"],"isi":["001523520000012"]},"year":"2025","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"_id":"34d33d68-11ca-11ed-8bc3-ec13763c0ca8","grant_number":"26293","name":"The impact of deleterious mutations on small populations"}],"publisher":"Institute of Mathematical Statistics","page":"1913-1940","article_processing_charge":"No","date_published":"2025-06-01T00:00:00Z","type":"journal_article","date_updated":"2025-09-30T14:12:48Z","publication":"The Annals of Applied Probability","date_created":"2025-07-21T08:13:54Z","publication_identifier":{"issn":["1050-5164"]},"abstract":[{"text":"We prove upper bounds on the L∞-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the L∞-Wasserstein metric and the relative L∞-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher’s infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.","lang":"eng"}],"oa":1,"title":"L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","OA_type":"green","article_type":"original","OA_place":"repository","acknowledgement":"This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/F65 and the Austrian Academy of Science, DOC fellowship nr. 26293.","scopus_import":"1","citation":{"apa":"Khudiakova, K., Maas, J., & Pedrotti, F. (2025). L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/25-aap2162","ama":"Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. 2025;35(3):1913-1940. doi:10.1214/25-aap2162","ieee":"K. Khudiakova, J. Maas, and F. Pedrotti, “L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model,” The Annals of Applied Probability, vol. 35, no. 3. Institute of Mathematical Statistics, pp. 1913–1940, 2025.","ista":"Khudiakova K, Maas J, Pedrotti F. 2025. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. 35(3), 1913–1940.","short":"K. Khudiakova, J. Maas, F. Pedrotti, The Annals of Applied Probability 35 (2025) 1913–1940.","mla":"Khudiakova, Kseniia, et al. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” The Annals of Applied Probability, vol. 35, no. 3, Institute of Mathematical Statistics, 2025, pp. 1913–40, doi:10.1214/25-aap2162.","chicago":"Khudiakova, Kseniia, Jan Maas, and Francesco Pedrotti. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/25-aap2162."},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2402.04151","open_access":"1"}],"status":"public","oa_version":"Preprint","quality_controlled":"1","language":[{"iso":"eng"}],"arxiv":1,"publication_status":"published","department":[{"_id":"JaMa"}],"day":"01","doi":"10.1214/25-aap2162","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"17352"}]},"corr_author":"1","month":"06"},{"article_processing_charge":"No","date_published":"2025-08-10T00:00:00Z","article_number":"2502.15665","year":"2025","project":[{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"},{"name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"author":[{"last_name":"Brigati","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","full_name":"Brigati, Giovanni","first_name":"Giovanni"},{"first_name":"Jan","orcid":"0000-0002-0845-1338","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan"},{"full_name":"Quattrocchi, Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","last_name":"Quattrocchi","orcid":"0009-0000-9773-1931","first_name":"Filippo"}],"_id":"20569","external_id":{"arxiv":["2502.15665"]},"arxiv":1,"language":[{"iso":"eng"}],"oa_version":"Preprint","month":"08","corr_author":"1","related_material":{"record":[{"status":"public","id":"20563","relation":"dissertation_contains"}]},"doi":"10.48550/arXiv.2502.15665","ec_funded":1,"day":"10","publication_status":"draft","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"acknowledgement":"This work was partially inspired by an unpublished note from 2014 by Guillaume Carlier,\r\nJean Dolbeault, and Bruno Nazaret. GB deeply thanks Jean Dolbeault for proposing\r\nthis problem to him, guiding him into the subject, and sharing the aforementioned note.\r\nWe are grateful to Karthik Elamvazhuthi for making us aware of the work [20].\r\nThe work of GB has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement\r\nNo 101034413.\r\nJM and FQ gratefully acknowledge support from the Austrian Science Fund (FWF)\r\nproject 10.55776/F65.","status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2502.15665","open_access":"1"}],"citation":{"apa":"Brigati, G., Maas, J., & Quattrocchi, F. (n.d.). Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv. https://doi.org/10.48550/arXiv.2502.15665","ama":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv. doi:10.48550/arXiv.2502.15665","ieee":"G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures,” arXiv. .","ista":"Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv, 2502.15665.","short":"G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.).","mla":"Brigati, Giovanni, et al. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” ArXiv, 2502.15665, doi:10.48550/arXiv.2502.15665.","chicago":"Brigati, Giovanni, Jan Maas, and Filippo Quattrocchi. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2502.15665."},"keyword":["optimal transport","kinetic theory","second-order discrepancy","Vlasov equation","Wasserstein splines."],"oa":1,"OA_place":"repository","OA_type":"green","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures","date_updated":"2026-06-24T22:30:42Z","type":"preprint","abstract":[{"text":"This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves.","lang":"eng"}],"publication":"arXiv","date_created":"2025-10-28T13:12:08Z"},{"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces","article_type":"original","oa":1,"publication":"Transactions of the American Mathematical Society","date_created":"2024-06-16T22:01:06Z","publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"abstract":[{"lang":"eng","text":"This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn.\r\n."}],"type":"journal_article","date_updated":"2026-04-07T13:00:02Z","publication_status":"published","department":[{"_id":"JaMa"}],"doi":"10.1090/tran/9156","ec_funded":1,"day":"01","related_material":{"record":[{"id":"17336","relation":"dissertation_contains","status":"public"}]},"month":"06","language":[{"iso":"eng"}],"oa_version":"Preprint","quality_controlled":"1","arxiv":1,"citation":{"apa":"Dello Schiavo, L., Maas, J., & Pedrotti, F. (2024). Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/9156","ama":"Dello Schiavo L, Maas J, Pedrotti F. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. 2024;377(6):3779-3804. doi:10.1090/tran/9156","ieee":"L. Dello Schiavo, J. Maas, and F. Pedrotti, “Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces,” Transactions of the American Mathematical Society, vol. 377, no. 6. American Mathematical Society, pp. 3779–3804, 2024.","ista":"Dello Schiavo L, Maas J, Pedrotti F. 2024. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. 377(6), 3779–3804.","chicago":"Dello Schiavo, Lorenzo, Jan Maas, and Francesco Pedrotti. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” Transactions of the American Mathematical Society. American Mathematical Society, 2024. https://doi.org/10.1090/tran/9156.","mla":"Dello Schiavo, Lorenzo, et al. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” Transactions of the American Mathematical Society, vol. 377, no. 6, American Mathematical Society, 2024, pp. 3779–804, doi:10.1090/tran/9156.","short":"L. Dello Schiavo, J. Maas, F. Pedrotti, Transactions of the American Mathematical Society 377 (2024) 3779–3804."},"status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2304.05239","open_access":"1"}],"scopus_import":"1","acknowledgement":"The authors gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 716117). This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/ESP208. This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/F65","issue":"6","_id":"17143","external_id":{"isi":["001203273300001"],"arxiv":["2304.05239"]},"author":[{"id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","full_name":"Dello Schiavo, Lorenzo","last_name":"Dello Schiavo","first_name":"Lorenzo","orcid":"0000-0002-9881-6870"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas"},{"last_name":"Pedrotti","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","full_name":"Pedrotti, Francesco","first_name":"Francesco"}],"isi":1,"intvolume":" 377","volume":377,"date_published":"2024-06-01T00:00:00Z","page":"3779-3804","article_processing_charge":"No","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"},{"_id":"34dbf174-11ca-11ed-8bc3-afe9d43d4b9c","grant_number":"E208","name":"Configuration Spaces over Non-Smooth Spaces"}],"publisher":"American Mathematical Society","year":"2024"},{"_id":"18897","external_id":{"arxiv":["2305.14164"]},"alternative_title":["TMLR"],"author":[{"last_name":"Pedrotti","full_name":"Pedrotti, Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","first_name":"Francesco"},{"last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"first_name":"Marco","orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco","last_name":"Mondelli"}],"project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"},{"name":"Prix Lopez-Loretta 2019 - Marco Mondelli","_id":"059876FA-7A3F-11EA-A408-12923DDC885E"}],"ddc":["000"],"year":"2024","date_published":"2024-06-01T00:00:00Z","article_processing_charge":"No","abstract":[{"lang":"eng","text":"Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time T1 by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires T1→∞. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as T1 diverges; from a practical viewpoint, a large T1 increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time T1. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the L2 loss on the score approximation, which is the quantity minimized in practice."}],"publication_identifier":{"issn":["2835-8856"]},"date_created":"2025-01-27T12:18:05Z","publication":"Transactions on Machine Learning Research","file_date_updated":"2025-01-27T12:19:44Z","date_updated":"2025-04-15T08:31:35Z","type":"conference","OA_place":"publisher","OA_type":"gold","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Improved convergence of score-based diffusion models via prediction-correction","oa":1,"has_accepted_license":"1","status":"public","file":[{"file_name":"2024_TMLR_Pedrotti.pdf","file_id":"18898","date_created":"2025-01-27T12:19:44Z","creator":"dernst","file_size":780315,"date_updated":"2025-01-27T12:19:44Z","access_level":"open_access","success":1,"checksum":"76a1fd5afd8ee6f7ae0e5912d7dbf6b4","relation":"main_file","content_type":"application/pdf"}],"citation":{"ieee":"F. Pedrotti, J. Maas, and M. Mondelli, “Improved convergence of score-based diffusion models via prediction-correction,” in Transactions on Machine Learning Research, 2024.","ama":"Pedrotti F, Maas J, Mondelli M. Improved convergence of score-based diffusion models via prediction-correction. In: Transactions on Machine Learning Research. ; 2024.","apa":"Pedrotti, F., Maas, J., & Mondelli, M. (2024). Improved convergence of score-based diffusion models via prediction-correction. In Transactions on Machine Learning Research.","short":"F. Pedrotti, J. Maas, M. Mondelli, in:, Transactions on Machine Learning Research, 2024.","chicago":"Pedrotti, Francesco, Jan Maas, and Marco Mondelli. “Improved Convergence of Score-Based Diffusion Models via Prediction-Correction.” In Transactions on Machine Learning Research, 2024.","mla":"Pedrotti, Francesco, et al. “Improved Convergence of Score-Based Diffusion Models via Prediction-Correction.” Transactions on Machine Learning Research, 2024.","ista":"Pedrotti F, Maas J, Mondelli M. 2024. Improved convergence of score-based diffusion models via prediction-correction. Transactions on Machine Learning Research. , TMLR, ."},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"acknowledgement":"Francesco Pedrotti and Jan Maas acknowledge support by the Austrian Science Fund (FWF) project 10.55776/F65. Marco Mondelli acknowledges support by the 2019 Lopez-Loreta prize.\r\n","scopus_import":"1","corr_author":"1","month":"06","related_material":{"record":[{"relation":"earlier_version","id":"17350","status":"public"}]},"day":"01","publication_status":"published","department":[{"_id":"JaMa"},{"_id":"MaMo"}],"arxiv":1,"quality_controlled":"1","language":[{"iso":"eng"}],"oa_version":"Published Version"},{"citation":{"ama":"Maas J, Rademacher SAE, Titkos T, Virosztek D, eds. Optimal Transport on Quantum Structures. Vol 29. Cham: Springer Nature; 2024. doi:10.1007/978-3-031-50466-2","apa":"Maas, J., Rademacher, S. A. E., Titkos, T., & Virosztek, D. (Eds.). (2024). Optimal Transport on Quantum Structures (Vol. 29). Cham: Springer Nature. https://doi.org/10.1007/978-3-031-50466-2","ieee":"J. Maas, S. A. E. Rademacher, T. Titkos, and D. Virosztek, Eds., Optimal Transport on Quantum Structures, vol. 29. Cham: Springer Nature, 2024.","ista":"Maas J, Rademacher SAE, Titkos T, Virosztek D eds. 2024. Optimal Transport on Quantum Structures, Cham: Springer Nature,p.","mla":"Maas, Jan, et al., editors. Optimal Transport on Quantum Structures. Vol. 29, Springer Nature, 2024, doi:10.1007/978-3-031-50466-2.","short":"J. Maas, S.A.E. Rademacher, T. Titkos, D. Virosztek, eds., Optimal Transport on Quantum Structures, Springer Nature, Cham, 2024.","chicago":"Maas, Jan, Simone Anna Elvira Rademacher, Tamás Titkos, and Daniel Virosztek, eds. Optimal Transport on Quantum Structures. Vol. 29. BSMS. Cham: Springer Nature, 2024. https://doi.org/10.1007/978-3-031-50466-2."},"publisher":"Springer Nature","status":"public","scopus_import":"1","year":"2024","day":"19","doi":"10.1007/978-3-031-50466-2","department":[{"_id":"JaMa"}],"publication_status":"published","month":"09","date_published":"2024-09-19T00:00:00Z","editor":[{"full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"first_name":"Simone Anna Elvira","orcid":"0000-0001-5059-4466","id":"856966FE-A408-11E9-977E-802DE6697425","full_name":"Rademacher, Simone Anna Elvira","last_name":"Rademacher"},{"first_name":"Tamás","full_name":"Titkos, Tamás","last_name":"Titkos"},{"first_name":"Daniel","orcid":"0000-0003-1109-5511","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel","last_name":"Virosztek"}],"quality_controlled":"1","oa_version":"None","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_identifier":{"eisbn":["9783031504662"],"eissn":["2947-9460"],"isbn":["9783031504655"],"issn":["1217-4696"]},"date_created":"2025-01-27T12:26:03Z","intvolume":" 29","abstract":[{"lang":"eng","text":"The flourishing theory of classical optimal transport concerns mass transportation at minimal cost. This book introduces the reader to optimal transport on quantum structures, i.e., optimal transportation between quantum states and related non-commutative concepts of mass transportation. It contains lecture notes on\r\n\r\nclassical optimal transport and Wasserstein gradient flows\r\ndynamics and quantum optimal transport\r\nquantum couplings and many-body problems\r\nquantum channels and qubits\r\n\r\nThese notes are based on lectures given by the authors at the \"Optimal Transport on Quantum Structures\" School held at the Erdös Center in Budapest in the fall of 2022. The lecture notes are complemented by two survey chapters presenting the state of the art in different research areas of non-commutative optimal transport."}],"series_title":"BSMS","date_updated":"2025-02-17T12:22:18Z","type":"book_editor","volume":29,"_id":"18899","alternative_title":["Bolyai Society Mathematical Studies"],"title":"Optimal Transport on Quantum Structures","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","place":"Cham"},{"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"},{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"publisher":"Springer Nature","ddc":["510"],"year":"2024","article_number":"153","date_published":"2024-07-01T00:00:00Z","pmid":1,"article_processing_charge":"Yes (via OA deal)","intvolume":" 63","volume":63,"external_id":{"pmid":["38947856"],"arxiv":["2209.11149"],"isi":["001258097800003"]},"_id":"17282","issue":"6","isi":1,"author":[{"id":"B7ECF9FC-AA38-11E9-AC9A-0930E6697425","full_name":"Brooks, Morris","last_name":"Brooks","first_name":"Morris","orcid":"0000-0002-6249-0928"},{"last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","first_name":"Jan"}],"status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"file_size":416622,"creator":"dernst","date_updated":"2024-07-22T07:05:32Z","file_name":"2024_CalculusVariations_Brooks.pdf","date_created":"2024-07-22T07:05:32Z","file_id":"17289","relation":"main_file","checksum":"a0cf0e0ba2157aabb287cb597be17dac","content_type":"application/pdf","success":1,"access_level":"open_access"}],"citation":{"ista":"Brooks M, Maas J. 2024. Characterisation of gradient flows for a given functional. Calculus of Variations and Partial Differential Equations. 63(6), 153.","chicago":"Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a given Functional.” Calculus of Variations and Partial Differential Equations. Springer Nature, 2024. https://doi.org/10.1007/s00526-024-02755-z.","mla":"Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a given Functional.” Calculus of Variations and Partial Differential Equations, vol. 63, no. 6, 153, Springer Nature, 2024, doi:10.1007/s00526-024-02755-z.","short":"M. Brooks, J. Maas, Calculus of Variations and Partial Differential Equations 63 (2024).","ama":"Brooks M, Maas J. Characterisation of gradient flows for a given functional. Calculus of Variations and Partial Differential Equations. 2024;63(6). doi:10.1007/s00526-024-02755-z","apa":"Brooks, M., & Maas, J. (2024). Characterisation of gradient flows for a given functional. Calculus of Variations and Partial Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-024-02755-z","ieee":"M. Brooks and J. Maas, “Characterisation of gradient flows for a given functional,” Calculus of Variations and Partial Differential Equations, vol. 63, no. 6. Springer Nature, 2024."},"scopus_import":"1","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).J. M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117), and by the Austrian Science Fund (FWF), Project SFB F65. We thank the anonymous referee for valuable comments on the paper.","corr_author":"1","month":"07","department":[{"_id":"JaMa"}],"publication_status":"published","day":"01","doi":"10.1007/s00526-024-02755-z","ec_funded":1,"arxiv":1,"language":[{"iso":"eng"}],"oa_version":"Published Version","quality_controlled":"1","abstract":[{"text":"Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ on M such that Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.","lang":"eng"}],"date_created":"2024-07-21T22:01:01Z","publication":"Calculus of Variations and Partial Differential Equations","publication_identifier":{"issn":["0944-2669"],"eissn":["1432-0835"]},"file_date_updated":"2024-07-22T07:05:32Z","type":"journal_article","date_updated":"2025-09-08T08:24:51Z","article_type":"original","title":"Characterisation of gradient flows for a given functional","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"has_accepted_license":"1"},{"date_updated":"2026-04-07T13:00:02Z","type":"preprint","abstract":[{"lang":"eng","text":"Score-based generative models (SGMs) are powerful tools to sample from\r\ncomplex data distributions. Their underlying idea is to (i) run a forward\r\nprocess for time $T_1$ by adding noise to the data, (ii) estimate its score\r\nfunction, and (iii) use such estimate to run a reverse process. As the reverse\r\nprocess is initialized with the stationary distribution of the forward one, the\r\nexisting analysis paradigm requires $T_1\\to\\infty$. This is however\r\nproblematic: from a theoretical viewpoint, for a given precision of the score\r\napproximation, the convergence guarantee fails as $T_1$ diverges; from a\r\npractical viewpoint, a large $T_1$ increases computational costs and leads to\r\nerror propagation. This paper addresses the issue by considering a version of\r\nthe popular predictor-corrector scheme: after running the forward process, we\r\nfirst estimate the final distribution via an inexact Langevin dynamics and then\r\nrevert the process. Our key technical contribution is to provide convergence\r\nguarantees which require to run the forward process only for a fixed finite\r\ntime $T_1$. Our bounds exhibit a mild logarithmic dependence on the input\r\ndimension and the subgaussian norm of the target distribution, have minimal\r\nassumptions on the data, and require only to control the $L^2$ loss on the\r\nscore approximation, which is the quantity minimized in practice."}],"date_created":"2024-07-31T07:56:40Z","publication":"arXiv","author":[{"last_name":"Pedrotti","full_name":"Pedrotti, Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","first_name":"Francesco"},{"full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"first_name":"Marco","orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco","last_name":"Mondelli"}],"oa":1,"OA_place":"repository","_id":"17350","external_id":{"arxiv":["2305.14164"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Improved convergence of score-based diffusion models via prediction-correction","year":"2024","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"_id":"059876FA-7A3F-11EA-A408-12923DDC885E","name":"Prix Lopez-Loretta 2019 - Marco Mondelli"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2305.14164"}],"status":"public","citation":{"ista":"Pedrotti F, Maas J, Mondelli M. Improved convergence of score-based diffusion models via prediction-correction. arXiv, 10.48550/arXiv.2305.14164.","mla":"Pedrotti, Francesco, et al. “Improved Convergence of Score-Based Diffusion Models via Prediction-Correction.” ArXiv, doi:10.48550/arXiv.2305.14164.","chicago":"Pedrotti, Francesco, Jan Maas, and Marco Mondelli. “Improved Convergence of Score-Based Diffusion Models via Prediction-Correction.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2305.14164.","short":"F. Pedrotti, J. Maas, M. Mondelli, ArXiv (n.d.).","apa":"Pedrotti, F., Maas, J., & Mondelli, M. (n.d.). Improved convergence of score-based diffusion models via prediction-correction. arXiv. https://doi.org/10.48550/arXiv.2305.14164","ama":"Pedrotti F, Maas J, Mondelli M. Improved convergence of score-based diffusion models via prediction-correction. arXiv. doi:10.48550/arXiv.2305.14164","ieee":"F. Pedrotti, J. Maas, and M. Mondelli, “Improved convergence of score-based diffusion models via prediction-correction,” arXiv. ."},"article_processing_charge":"No","arxiv":1,"oa_version":"Preprint","language":[{"iso":"eng"}],"month":"06","date_published":"2024-06-06T00:00:00Z","corr_author":"1","related_material":{"record":[{"status":"public","relation":"later_version","id":"18897"},{"status":"public","id":"17336","relation":"dissertation_contains"}]},"day":"06","doi":"10.48550/arXiv.2305.14164","publication_status":"draft","department":[{"_id":"JaMa"},{"_id":"MaMo"}]},{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2402.04151","open_access":"1"}],"project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"},{"grant_number":"26293","name":"The impact of deleterious mutations on small populations","_id":"34d33d68-11ca-11ed-8bc3-ec13763c0ca8"}],"status":"public","citation":{"ista":"Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv, 2402.04151.","short":"K. Khudiakova, J. Maas, F. Pedrotti, ArXiv (n.d.).","mla":"Khudiakova, Kseniia, et al. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” ArXiv, 2402.04151, doi:10.48550/arXiv.2402.04151.","chicago":"Khudiakova, Kseniia, Jan Maas, and Francesco Pedrotti. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2402.04151.","apa":"Khudiakova, K., Maas, J., & Pedrotti, F. (n.d.). L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv. https://doi.org/10.48550/arXiv.2402.04151","ama":"Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv. doi:10.48550/arXiv.2402.04151","ieee":"K. Khudiakova, J. Maas, and F. Pedrotti, “L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model,” arXiv. ."},"year":"2024","article_number":"2402.04151","related_material":{"record":[{"status":"public","relation":"later_version","id":"20050"},{"status":"public","relation":"dissertation_contains","id":"17336"}]},"date_published":"2024-02-07T00:00:00Z","corr_author":"1","month":"02","department":[{"_id":"JaMa"}],"publication_status":"draft","day":"07","doi":"10.48550/arXiv.2402.04151","arxiv":1,"article_processing_charge":"No","language":[{"iso":"eng"}],"oa_version":"Preprint","abstract":[{"text":"We prove upper bounds on the $L^\\infty$-Wasserstein distance from optimal\r\ntransport between strongly log-concave probability densities and log-Lipschitz\r\nperturbations. In the simplest setting, such a bound amounts to a\r\ntransport-information inequality involving the $L^\\infty$-Wasserstein metric\r\nand the relative $L^\\infty$-Fisher information. We show that this inequality\r\ncan be sharpened significantly in situations where the involved densities are\r\nanisotropic. Our proof is based on probabilistic techniques using Langevin\r\ndynamics. As an application of these results, we obtain sharp exponential rates\r\nof convergence in Fisher's infinitesimal model from quantitative genetics,\r\ngeneralising recent results by Calvez, Poyato, and Santambrogio in dimension 1\r\nto arbitrary dimensions.","lang":"eng"}],"publication":"arXiv","date_created":"2024-07-31T08:07:40Z","type":"preprint","date_updated":"2026-04-07T13:00:02Z","OA_place":"repository","title":"L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"17352","external_id":{"arxiv":["2402.04151"]},"author":[{"last_name":"Khudiakova","full_name":"Khudiakova, Kseniia","id":"4E6DC800-AE37-11E9-AC72-31CAE5697425","orcid":"0000-0002-6246-1465","first_name":"Kseniia"},{"last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"first_name":"Francesco","last_name":"Pedrotti","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","full_name":"Pedrotti, Francesco"}],"oa":1},{"language":[{"iso":"eng"}],"quality_controlled":"1","oa_version":"Published Version","arxiv":1,"publication_status":"published","department":[{"_id":"JaMa"}],"ec_funded":1,"doi":"10.1007/s00526-023-02472-z","day":"28","corr_author":"1","month":"04","acknowledgement":"J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117). J.M and L.P. also acknowledge support from the Austrian Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—350398276. We thank the anonymous reviewer for the careful reading and for useful suggestions. Open access funding provided by Austrian Science Fund (FWF).","scopus_import":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"citation":{"apa":"Gladbach, P., Kopfer, E., Maas, J., & Portinale, L. (2023). Homogenisation of dynamical optimal transport on periodic graphs. Calculus of Variations and Partial Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-023-02472-z","ama":"Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of dynamical optimal transport on periodic graphs. Calculus of Variations and Partial Differential Equations. 2023;62(5). doi:10.1007/s00526-023-02472-z","ieee":"P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of dynamical optimal transport on periodic graphs,” Calculus of Variations and Partial Differential Equations, vol. 62, no. 5. Springer Nature, 2023.","ista":"Gladbach P, Kopfer E, Maas J, Portinale L. 2023. Homogenisation of dynamical optimal transport on periodic graphs. Calculus of Variations and Partial Differential Equations. 62(5), 143.","chicago":"Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” Calculus of Variations and Partial Differential Equations. Springer Nature, 2023. https://doi.org/10.1007/s00526-023-02472-z.","short":"P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and Partial Differential Equations 62 (2023).","mla":"Gladbach, Peter, et al. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” Calculus of Variations and Partial Differential Equations, vol. 62, no. 5, 143, Springer Nature, 2023, doi:10.1007/s00526-023-02472-z."},"file":[{"success":1,"access_level":"open_access","relation":"main_file","checksum":"359bee38d94b7e0aa73925063cb8884d","content_type":"application/pdf","file_name":"2023_CalculusEquations_Gladbach.pdf","file_id":"14393","date_created":"2023-10-04T11:34:10Z","file_size":1240995,"creator":"dernst","date_updated":"2023-10-04T11:34:10Z"}],"status":"public","has_accepted_license":"1","oa":1,"title":"Homogenisation of dynamical optimal transport on periodic graphs","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_type":"original","type":"journal_article","date_updated":"2025-05-15T10:54:12Z","file_date_updated":"2023-10-04T11:34:10Z","date_created":"2023-05-14T22:01:00Z","publication":"Calculus of Variations and Partial Differential Equations","publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"abstract":[{"lang":"eng","text":"This paper deals with the large-scale behaviour of dynamical optimal transport on Zd\r\n-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ\r\n-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs."}],"article_processing_charge":"Yes (via OA deal)","date_published":"2023-04-28T00:00:00Z","pmid":1,"year":"2023","article_number":"143","ddc":["510"],"project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"},{"grant_number":"W1245","name":"Dissipation and dispersion in nonlinear partial differential equations","call_identifier":"FWF","_id":"260788DE-B435-11E9-9278-68D0E5697425"}],"publisher":"Springer Nature","isi":1,"author":[{"last_name":"Gladbach","full_name":"Gladbach, Peter","first_name":"Peter"},{"full_name":"Kopfer, Eva","last_name":"Kopfer","first_name":"Eva"},{"full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","full_name":"Portinale, Lorenzo","last_name":"Portinale","first_name":"Lorenzo"}],"external_id":{"isi":["000980588900001"],"arxiv":["2110.15321"],"pmid":["37131846"]},"_id":"12959","issue":"5","volume":62,"intvolume":" 62"},{"year":"2022","publisher":"American Institute of Mathematical Sciences","project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"page":"687-717","article_processing_charge":"No","date_published":"2022-10-01T00:00:00Z","volume":17,"intvolume":" 17","isi":1,"author":[{"last_name":"Erbar","full_name":"Erbar, Matthias","first_name":"Matthias"},{"full_name":"Forkert, Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87","last_name":"Forkert","first_name":"Dominik L"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas"},{"first_name":"Delio","full_name":"Mugnolo, Delio","last_name":"Mugnolo"}],"_id":"11700","external_id":{"isi":["000812422100001"],"arxiv":["2105.05677"]},"issue":"5","acknowledgement":"ME acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG), Grant SFB 1283/2 2021 – 317210226. DF and JM were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117). JM also acknowledges support by the Austrian Science Fund (FWF), Project SFB F65. The work of DM was partially supported by the Deutsche Forschungsgemeinschaft\r\n(DFG), Grant 397230547. This article is based upon work from COST Action\r\n18232 MAT-DYN-NET, supported by COST (European Cooperation in Science\r\nand Technology), www.cost.eu. We wish to thank Martin Burger and Jan-Frederik\r\nPietschmann for useful discussions. We are grateful to the anonymous referees for\r\ntheir careful reading and useful suggestions.","scopus_import":"1","citation":{"apa":"Erbar, M., Forkert, D. L., Maas, J., & Mugnolo, D. (2022). Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media. American Institute of Mathematical Sciences. https://doi.org/10.3934/nhm.2022023","ama":"Erbar M, Forkert DL, Maas J, Mugnolo D. Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media. 2022;17(5):687-717. doi:10.3934/nhm.2022023","ieee":"M. Erbar, D. L. Forkert, J. Maas, and D. Mugnolo, “Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph,” Networks and Heterogeneous Media, vol. 17, no. 5. American Institute of Mathematical Sciences, pp. 687–717, 2022.","ista":"Erbar M, Forkert DL, Maas J, Mugnolo D. 2022. Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media. 17(5), 687–717.","mla":"Erbar, Matthias, et al. “Gradient Flow Formulation of Diffusion Equations in the Wasserstein Space over a Metric Graph.” Networks and Heterogeneous Media, vol. 17, no. 5, American Institute of Mathematical Sciences, 2022, pp. 687–717, doi:10.3934/nhm.2022023.","short":"M. Erbar, D.L. Forkert, J. Maas, D. Mugnolo, Networks and Heterogeneous Media 17 (2022) 687–717.","chicago":"Erbar, Matthias, Dominik L Forkert, Jan Maas, and Delio Mugnolo. “Gradient Flow Formulation of Diffusion Equations in the Wasserstein Space over a Metric Graph.” Networks and Heterogeneous Media. American Institute of Mathematical Sciences, 2022. https://doi.org/10.3934/nhm.2022023."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2105.05677"}],"status":"public","oa_version":"Preprint","language":[{"iso":"eng"}],"quality_controlled":"1","arxiv":1,"day":"01","ec_funded":1,"doi":"10.3934/nhm.2022023","department":[{"_id":"JaMa"}],"publication_status":"published","month":"10","corr_author":"1","date_updated":"2025-04-14T07:27:47Z","type":"journal_article","publication_identifier":{"eissn":["1556-181X"],"issn":["1556-1801"]},"publication":"Networks and Heterogeneous Media","date_created":"2022-07-31T22:01:46Z","abstract":[{"lang":"eng","text":"This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space."}],"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph","article_type":"original"},{"volume":54,"intvolume":" 54","isi":1,"author":[{"first_name":"Dominik L","full_name":"Forkert, Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87","last_name":"Forkert"},{"last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"first_name":"Lorenzo","last_name":"Portinale","full_name":"Portinale, Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87"}],"external_id":{"arxiv":["2008.10962"],"isi":["000889274600001"]},"_id":"11739","issue":"4","year":"2022","publisher":"Society for Industrial and Applied Mathematics","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","grant_number":"W1245","call_identifier":"FWF","name":"Dissipation and dispersion in nonlinear partial differential equations"}],"article_processing_charge":"No","page":"4297-4333","date_published":"2022-07-18T00:00:00Z","date_updated":"2025-04-15T08:31:31Z","type":"journal_article","abstract":[{"text":"We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.","lang":"eng"}],"publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"publication":"SIAM Journal on Mathematical Analysis","date_created":"2022-08-07T22:01:59Z","keyword":["Fokker--Planck equation","gradient flow","evolutionary $\\Gamma$-convergence"],"oa":1,"article_type":"original","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","title":"Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","scopus_import":"1","acknowledgement":"This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme grant 716117 and by the AustrianScience Fund (FWF) through grants F65 and W1245.","status":"public","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2008.10962"}],"citation":{"ieee":"D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” SIAM Journal on Mathematical Analysis, vol. 54, no. 4. Society for Industrial and Applied Mathematics, pp. 4297–4333, 2022.","ama":"Forkert DL, Maas J, Portinale L. Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. 2022;54(4):4297-4333. doi:10.1137/21M1410968","apa":"Forkert, D. L., Maas, J., & Portinale, L. (2022). Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21M1410968","chicago":"Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21M1410968.","mla":"Forkert, Dominik L., et al. “Evolutionary $\\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial and Applied Mathematics, 2022, pp. 4297–333, doi:10.1137/21M1410968.","short":"D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333.","ista":"Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. 54(4), 4297–4333."},"arxiv":1,"oa_version":"Preprint","language":[{"iso":"eng"}],"quality_controlled":"1","month":"07","corr_author":"1","related_material":{"record":[{"relation":"earlier_version","id":"10022","status":"public"}]},"ec_funded":1,"doi":"10.1137/21M1410968","day":"18","publication_status":"published","department":[{"_id":"JaMa"}]},{"intvolume":" 21","volume":21,"external_id":{"arxiv":["2005.14177"]},"_id":"10023","issue":"4","author":[{"last_name":"Karatzas","full_name":"Karatzas, Ioannis","first_name":"Ioannis"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Schachermayer, Walter","last_name":"Schachermayer","first_name":"Walter"}],"project":[{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"publisher":"International Press","year":"2021","date_published":"2021-06-04T00:00:00Z","article_processing_charge":"No","page":"481-536","abstract":[{"text":"We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.","lang":"eng"}],"publication":"Communications in Information and Systems","date_created":"2021-09-19T08:53:19Z","publication_identifier":{"issn":["1526-7555"]},"type":"journal_article","date_updated":"2025-04-14T07:27:45Z","article_type":"original","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","title":"Trajectorial dissipation and gradient flow for the relative entropy in Markov chains","keyword":["Markov Chain","relative entropy","time reversal","steepest descent","gradient flow"],"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2005.14177"}],"status":"public","citation":{"apa":"Karatzas, I., Maas, J., & Schachermayer, W. (2021). Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. Communications in Information and Systems. International Press. https://doi.org/10.4310/CIS.2021.v21.n4.a1","ama":"Karatzas I, Maas J, Schachermayer W. Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. Communications in Information and Systems. 2021;21(4):481-536. doi:10.4310/CIS.2021.v21.n4.a1","ieee":"I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and gradient flow for the relative entropy in Markov chains,” Communications in Information and Systems, vol. 21, no. 4. International Press, pp. 481–536, 2021.","ista":"Karatzas I, Maas J, Schachermayer W. 2021. Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. Communications in Information and Systems. 21(4), 481–536.","mla":"Karatzas, Ioannis, et al. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” Communications in Information and Systems, vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:10.4310/CIS.2021.v21.n4.a1.","chicago":"Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” Communications in Information and Systems. International Press, 2021. https://doi.org/10.4310/CIS.2021.v21.n4.a1.","short":"I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and Systems 21 (2021) 481–536."},"acknowledgement":"I.K. acknowledges support from the U.S. National Science Foundation under Grant NSF-DMS-20-04997. J.M. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117) and from the Austrian Science Fund (FWF) through project F65. W.S. acknowledges support from the Austrian Science Fund (FWF) under grant P28861 and by the Vienna Science and Technology Fund (WWTF) through projects MA14-008 and MA16-021.","month":"06","publication_status":"published","department":[{"_id":"JaMa"}],"day":"04","doi":"10.4310/CIS.2021.v21.n4.a1","ec_funded":1,"arxiv":1,"quality_controlled":"1","oa_version":"Preprint","language":[{"iso":"eng"}]},{"_id":"6358","issue":"2","external_id":{"pmid":["33223567"],"arxiv":["1811.04572"],"isi":["000498933300001"]},"isi":1,"author":[{"first_name":"Eric A.","last_name":"Carlen","full_name":"Carlen, Eric A."},{"full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","first_name":"Jan"}],"intvolume":" 178","volume":178,"date_published":"2020-01-01T00:00:00Z","pmid":1,"page":"319-378","article_processing_charge":"Yes (via OA deal)","ddc":["500"],"publisher":"Springer Nature","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"},{"grant_number":"F06504","call_identifier":"FWF","name":"Taming Complexity in Partial Differential Systems","_id":"260482E2-B435-11E9-9278-68D0E5697425"}],"year":"2020","title":"Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_type":"original","oa":1,"has_accepted_license":"1","publication_identifier":{"eissn":["1572-9613"],"issn":["0022-4715"]},"publication":"Journal of Statistical Physics","date_created":"2019-04-30T07:34:18Z","abstract":[{"lang":"eng","text":"We study dynamical optimal transport metrics between density matricesassociated to symmetric Dirichlet forms on finite-dimensional C∗-algebras. Our settingcovers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, andspectral gap estimates."}],"date_updated":"2025-06-12T07:27:20Z","type":"journal_article","file_date_updated":"2020-07-14T12:47:28Z","doi":"10.1007/s10955-019-02434-w","ec_funded":1,"day":"01","department":[{"_id":"JaMa"}],"publication_status":"published","corr_author":"1","month":"01","related_material":{"link":[{"url":"https://doi.org/10.1007/s10955-020-02671-4","relation":"erratum"}]},"quality_controlled":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"arxiv":1,"citation":{"apa":"Carlen, E. A., & Maas, J. (2020). Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. Journal of Statistical Physics. Springer Nature. https://doi.org/10.1007/s10955-019-02434-w","ama":"Carlen EA, Maas J. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. Journal of Statistical Physics. 2020;178(2):319-378. doi:10.1007/s10955-019-02434-w","ieee":"E. A. Carlen and J. Maas, “Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems,” Journal of Statistical Physics, vol. 178, no. 2. Springer Nature, pp. 319–378, 2020.","ista":"Carlen EA, Maas J. 2020. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. Journal of Statistical Physics. 178(2), 319–378.","mla":"Carlen, Eric A., and Jan Maas. “Non-Commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems.” Journal of Statistical Physics, vol. 178, no. 2, Springer Nature, 2020, pp. 319–78, doi:10.1007/s10955-019-02434-w.","chicago":"Carlen, Eric A., and Jan Maas. “Non-Commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems.” Journal of Statistical Physics. Springer Nature, 2020. https://doi.org/10.1007/s10955-019-02434-w.","short":"E.A. Carlen, J. Maas, Journal of Statistical Physics 178 (2020) 319–378."},"file":[{"relation":"main_file","checksum":"7b04befbdc0d4982c0ee945d25d19872","content_type":"application/pdf","access_level":"open_access","file_size":905538,"creator":"dernst","date_updated":"2020-07-14T12:47:28Z","file_name":"2019_JourStatistPhysics_Carlen.pdf","date_created":"2019-12-23T12:03:09Z","file_id":"7209"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"status":"public","scopus_import":"1"},{"_id":"10022","external_id":{"arxiv":["2008.10962"]},"author":[{"first_name":"Dominik L","id":"35C79D68-F248-11E8-B48F-1D18A9856A87","full_name":"Forkert, Dominik L","last_name":"Forkert"},{"first_name":"Jan","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","last_name":"Maas"},{"full_name":"Portinale, Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","last_name":"Portinale","first_name":"Lorenzo"}],"date_published":"2020-08-25T00:00:00Z","article_processing_charge":"No","project":[{"grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"year":"2020","article_number":"2008.10962","title":"Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"abstract":[{"text":"We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in R^d and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.","lang":"eng"}],"publication":"arXiv","date_created":"2021-09-17T10:57:27Z","type":"preprint","date_updated":"2026-04-08T07:00:03Z","related_material":{"record":[{"relation":"later_version","id":"11739","status":"public"},{"status":"public","relation":"dissertation_contains","id":"10030"}]},"month":"08","corr_author":"1","publication_status":"draft","department":[{"_id":"JaMa"}],"ec_funded":1,"day":"25","doi":"10.48550/arXiv.2008.10962","arxiv":1,"oa_version":"Preprint","language":[{"iso":"eng"}],"status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2008.10962"}],"citation":{"apa":"Forkert, D. L., Maas, J., & Portinale, L. (n.d.). Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv. https://doi.org/10.48550/arXiv.2008.10962","ama":"Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv. doi:10.48550/arXiv.2008.10962","ieee":"D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” arXiv. .","ista":"Forkert DL, Maas J, Portinale L. Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. arXiv, 2008.10962.","mla":"Forkert, Dominik L., et al. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” ArXiv, 2008.10962, doi:10.48550/arXiv.2008.10962.","chicago":"Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary Γ-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2008.10962.","short":"D.L. Forkert, J. Maas, L. Portinale, ArXiv (n.d.)."},"acknowledgement":"This work is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117) and by the Austrian Science Fund (FWF), grants No F65 and W1245."},{"date_published":"2020-07-01T00:00:00Z","page":"204-234","article_processing_charge":"No","publisher":"Elsevier","project":[{"name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Taming Complexity in Partial Differential Systems","grant_number":"F06504"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","grant_number":"W1245","call_identifier":"FWF","name":"Dissipation and dispersion in nonlinear partial differential equations"}],"year":"2020","external_id":{"isi":["000539439400008"],"arxiv":["1905.05757"]},"_id":"7573","issue":"7","author":[{"full_name":"Gladbach, Peter","last_name":"Gladbach","first_name":"Peter"},{"last_name":"Kopfer","full_name":"Kopfer, Eva","first_name":"Eva"},{"last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338","first_name":"Jan"},{"first_name":"Lorenzo","full_name":"Portinale, Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","last_name":"Portinale"}],"isi":1,"intvolume":" 139","volume":139,"ec_funded":1,"day":"01","doi":"10.1016/j.matpur.2020.02.008","department":[{"_id":"JaMa"}],"publication_status":"published","month":"07","related_material":{"record":[{"id":"10030","relation":"dissertation_contains","status":"public"}]},"language":[{"iso":"eng"}],"oa_version":"Preprint","quality_controlled":"1","arxiv":1,"citation":{"short":"P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Journal de Mathematiques Pures et Appliquees 139 (2020) 204–234.","mla":"Gladbach, Peter, et al. “Homogenisation of One-Dimensional Discrete Optimal Transport.” Journal de Mathematiques Pures et Appliquees, vol. 139, no. 7, Elsevier, 2020, pp. 204–34, doi:10.1016/j.matpur.2020.02.008.","chicago":"Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation of One-Dimensional Discrete Optimal Transport.” Journal de Mathematiques Pures et Appliquees. Elsevier, 2020. https://doi.org/10.1016/j.matpur.2020.02.008.","ista":"Gladbach P, Kopfer E, Maas J, Portinale L. 2020. Homogenisation of one-dimensional discrete optimal transport. Journal de Mathematiques Pures et Appliquees. 139(7), 204–234.","ieee":"P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of one-dimensional discrete optimal transport,” Journal de Mathematiques Pures et Appliquees, vol. 139, no. 7. Elsevier, pp. 204–234, 2020.","apa":"Gladbach, P., Kopfer, E., Maas, J., & Portinale, L. (2020). Homogenisation of one-dimensional discrete optimal transport. Journal de Mathematiques Pures et Appliquees. Elsevier. https://doi.org/10.1016/j.matpur.2020.02.008","ama":"Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of one-dimensional discrete optimal transport. Journal de Mathematiques Pures et Appliquees. 2020;139(7):204-234. doi:10.1016/j.matpur.2020.02.008"},"main_file_link":[{"url":"https://arxiv.org/abs/1905.05757","open_access":"1"}],"status":"public","scopus_import":"1","acknowledgement":"J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 716117). J.M. and L.P. also acknowledge support from the Austrian Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 350398276.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Homogenisation of one-dimensional discrete optimal transport","article_type":"original","oa":1,"publication_identifier":{"issn":["0021-7824"]},"publication":"Journal de Mathematiques Pures et Appliquees","date_created":"2020-03-08T23:00:47Z","abstract":[{"lang":"eng","text":"This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such metrics appear naturally in discretisations of -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport."}],"date_updated":"2026-04-08T07:00:03Z","type":"journal_article"},{"abstract":[{"text":"We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.","lang":"eng"}],"publication_identifier":{"eissn":["1572-9613"],"issn":["0022-4715"]},"date_created":"2020-11-15T23:01:18Z","publication":"Journal of Statistical Physics","file_date_updated":"2021-02-04T10:29:11Z","date_updated":"2025-06-12T07:01:39Z","type":"journal_article","article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Modeling of chemical reaction systems with detailed balance using gradient structures","oa":1,"has_accepted_license":"1","status":"public","citation":{"short":"J. Maas, A. Mielke, Journal of Statistical Physics 181 (2020) 2257–2303.","mla":"Maas, Jan, and Alexander Mielke. “Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures.” Journal of Statistical Physics, vol. 181, no. 6, Springer Nature, 2020, pp. 2257–303, doi:10.1007/s10955-020-02663-4.","chicago":"Maas, Jan, and Alexander Mielke. “Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures.” Journal of Statistical Physics. Springer Nature, 2020. https://doi.org/10.1007/s10955-020-02663-4.","ista":"Maas J, Mielke A. 2020. Modeling of chemical reaction systems with detailed balance using gradient structures. Journal of Statistical Physics. 181(6), 2257–2303.","ieee":"J. Maas and A. Mielke, “Modeling of chemical reaction systems with detailed balance using gradient structures,” Journal of Statistical Physics, vol. 181, no. 6. Springer Nature, pp. 2257–2303, 2020.","apa":"Maas, J., & Mielke, A. (2020). Modeling of chemical reaction systems with detailed balance using gradient structures. Journal of Statistical Physics. Springer Nature. https://doi.org/10.1007/s10955-020-02663-4","ama":"Maas J, Mielke A. Modeling of chemical reaction systems with detailed balance using gradient structures. Journal of Statistical Physics. 2020;181(6):2257-2303. doi:10.1007/s10955-020-02663-4"},"file":[{"content_type":"application/pdf","checksum":"bc2b63a90197b97cbc73eccada4639f5","relation":"main_file","access_level":"open_access","success":1,"date_updated":"2021-02-04T10:29:11Z","creator":"dernst","file_size":753596,"file_id":"9087","date_created":"2021-02-04T10:29:11Z","file_name":"2020_JourStatPhysics_Maas.pdf"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"scopus_import":"1","acknowledgement":"The research of A.M. was partially supported by the Deutsche Forschungsgemeinschaft (DFG) via the Collaborative Research Center SFB 1114 Scaling Cascades in Complex Systems (Project No. 235221301), through the Subproject C05 Effective models for materials and interfaces with multiple scales. J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117), and by the Austrian Science Fund (FWF), Project SFB F65. The authors thank Christof Schütte, Robert I. A. Patterson, and Stefanie Winkelmann for helpful and stimulating discussions. Open access funding provided by Austrian Science Fund (FWF).","corr_author":"1","month":"12","doi":"10.1007/s10955-020-02663-4","ec_funded":1,"day":"01","publication_status":"published","department":[{"_id":"JaMa"}],"arxiv":1,"oa_version":"Published Version","language":[{"iso":"eng"}],"quality_controlled":"1","intvolume":" 181","volume":181,"external_id":{"pmid":["33268907"],"isi":["000587107200002"],"arxiv":["2004.02831"]},"_id":"8758","issue":"6","author":[{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","last_name":"Maas","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"last_name":"Mielke","full_name":"Mielke, Alexander","first_name":"Alexander"}],"isi":1,"publisher":"Springer Nature","project":[{"grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"260482E2-B435-11E9-9278-68D0E5697425","name":"Taming Complexity in Partial Differential Systems","call_identifier":"FWF","grant_number":"F06504"}],"ddc":["510"],"year":"2020","date_published":"2020-12-01T00:00:00Z","pmid":1,"article_processing_charge":"No","page":"2257-2303"},{"year":"2020","publisher":"Society for Industrial and Applied Mathematics","article_processing_charge":"No","page":"2759-2802","date_published":"2020-10-01T00:00:00Z","volume":52,"intvolume":" 52","publist_id":"7983","author":[{"first_name":"Peter","last_name":"Gladbach","full_name":"Gladbach, Peter"},{"first_name":"Eva","last_name":"Kopfer","full_name":"Kopfer, Eva"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","last_name":"Maas","first_name":"Jan","orcid":"0000-0002-0845-1338"}],"isi":1,"issue":"3","_id":"71","external_id":{"arxiv":["1809.01092"],"isi":["000546975100017"]},"scopus_import":"1","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1809.01092"}],"citation":{"chicago":"Gladbach, Peter, Eva Kopfer, and Jan Maas. “Scaling Limits of Discrete Optimal Transport.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2020. https://doi.org/10.1137/19M1243440.","mla":"Gladbach, Peter, et al. “Scaling Limits of Discrete Optimal Transport.” SIAM Journal on Mathematical Analysis, vol. 52, no. 3, Society for Industrial and Applied Mathematics, 2020, pp. 2759–802, doi:10.1137/19M1243440.","short":"P. Gladbach, E. Kopfer, J. Maas, SIAM Journal on Mathematical Analysis 52 (2020) 2759–2802.","ista":"Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 2759–2802.","ieee":"P. Gladbach, E. Kopfer, and J. Maas, “Scaling limits of discrete optimal transport,” SIAM Journal on Mathematical Analysis, vol. 52, no. 3. Society for Industrial and Applied Mathematics, pp. 2759–2802, 2020.","apa":"Gladbach, P., Kopfer, E., & Maas, J. (2020). Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1243440","ama":"Gladbach P, Kopfer E, Maas J. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 2020;52(3):2759-2802. doi:10.1137/19M1243440"},"arxiv":1,"language":[{"iso":"eng"}],"quality_controlled":"1","oa_version":"Preprint","month":"10","doi":"10.1137/19M1243440","day":"01","department":[{"_id":"JaMa"}],"publication_status":"published","date_updated":"2025-07-10T11:54:14Z","type":"journal_article","abstract":[{"lang":"eng","text":"We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric."}],"publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"publication":"SIAM Journal on Mathematical Analysis","date_created":"2018-12-11T11:44:28Z","oa":1,"article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Scaling limits of discrete optimal transport"},{"date_published":"2019-02-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","project":[{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"grant_number":"F06504","name":"Taming Complexity in Partial Differential Systems","call_identifier":"FWF","_id":"260482E2-B435-11E9-9278-68D0E5697425"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"publisher":"Springer","ddc":["510"],"year":"2019","article_number":"19","issue":"1","_id":"73","external_id":{"arxiv":["1805.06040"],"isi":["000452849400001"]},"isi":1,"author":[{"first_name":"Matthias","full_name":"Erbar, Matthias","last_name":"Erbar"},{"last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"first_name":"Melchior","full_name":"Wirth, Melchior","last_name":"Wirth"}],"intvolume":" 58","volume":58,"month":"02","publication_status":"published","department":[{"_id":"JaMa"}],"day":"01","ec_funded":1,"doi":"10.1007/s00526-018-1456-1","arxiv":1,"language":[{"iso":"eng"}],"oa_version":"Published Version","quality_controlled":"1","status":"public","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file":[{"access_level":"open_access","content_type":"application/pdf","relation":"main_file","checksum":"ba05ac2d69de4c58d2cd338b63512798","file_id":"5895","date_created":"2019-01-28T15:37:11Z","file_name":"2018_Calculus_Erbar.pdf","date_updated":"2020-07-14T12:47:55Z","file_size":645565,"creator":"dernst"}],"citation":{"chicago":"Erbar, Matthias, Jan Maas, and Melchior Wirth. “On the Geometry of Geodesics in Discrete Optimal Transport.” Calculus of Variations and Partial Differential Equations. Springer, 2019. https://doi.org/10.1007/s00526-018-1456-1.","mla":"Erbar, Matthias, et al. “On the Geometry of Geodesics in Discrete Optimal Transport.” Calculus of Variations and Partial Differential Equations, vol. 58, no. 1, 19, Springer, 2019, doi:10.1007/s00526-018-1456-1.","short":"M. Erbar, J. Maas, M. Wirth, Calculus of Variations and Partial Differential Equations 58 (2019).","ista":"Erbar M, Maas J, Wirth M. 2019. On the geometry of geodesics in discrete optimal transport. Calculus of Variations and Partial Differential Equations. 58(1), 19.","ieee":"M. Erbar, J. Maas, and M. Wirth, “On the geometry of geodesics in discrete optimal transport,” Calculus of Variations and Partial Differential Equations, vol. 58, no. 1. Springer, 2019.","apa":"Erbar, M., Maas, J., & Wirth, M. (2019). On the geometry of geodesics in discrete optimal transport. Calculus of Variations and Partial Differential Equations. Springer. https://doi.org/10.1007/s00526-018-1456-1","ama":"Erbar M, Maas J, Wirth M. On the geometry of geodesics in discrete optimal transport. Calculus of Variations and Partial Differential Equations. 2019;58(1). doi:10.1007/s00526-018-1456-1"},"scopus_import":"1","article_type":"original","title":"On the geometry of geodesics in discrete optimal transport","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","has_accepted_license":"1","oa":1,"abstract":[{"lang":"eng","text":"We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest."}],"date_created":"2018-12-11T11:44:29Z","publication":"Calculus of Variations and Partial Differential Equations","publication_identifier":{"issn":["0944-2669"]},"file_date_updated":"2020-07-14T12:47:55Z","type":"journal_article","date_updated":"2026-04-16T09:51:42Z"},{"volume":273,"intvolume":" 273","publist_id":"6452","author":[{"full_name":"Carlen, Eric","last_name":"Carlen","first_name":"Eric"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas"}],"isi":1,"_id":"956","issue":"5","external_id":{"isi":["000406082300005"],"arxiv":["1609.01254"]},"year":"2017","publisher":"Academic Press","article_processing_charge":"No","page":"1810 - 1869","date_published":"2017-09-01T00:00:00Z","date_updated":"2025-06-04T08:14:53Z","type":"journal_article","abstract":[{"lang":"eng","text":"We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance."}],"publication_identifier":{"issn":["0022-1236"]},"publication":"Journal of Functional Analysis","date_created":"2018-12-11T11:49:24Z","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance","scopus_import":"1","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1609.01254"}],"citation":{"mla":"Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis, vol. 273, no. 5, Academic Press, 2017, pp. 1810–69, doi:10.1016/j.jfa.2017.05.003.","chicago":"Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum Markov Semigroups with Detailed Balance.” Journal of Functional Analysis. Academic Press, 2017. https://doi.org/10.1016/j.jfa.2017.05.003.","short":"E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869.","ista":"Carlen E, Maas J. 2017. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis. 273(5), 1810–1869.","ieee":"E. Carlen and J. Maas, “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance,” Journal of Functional Analysis, vol. 273, no. 5. Academic Press, pp. 1810–1869, 2017.","apa":"Carlen, E., & Maas, J. (2017). Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2017.05.003","ama":"Carlen E, Maas J. Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. Journal of Functional Analysis. 2017;273(5):1810-1869. doi:10.1016/j.jfa.2017.05.003"},"arxiv":1,"oa_version":"Submitted Version","quality_controlled":"1","language":[{"iso":"eng"}],"month":"09","day":"01","doi":"10.1016/j.jfa.2017.05.003","publication_status":"published","department":[{"_id":"JaMa"}]},{"doi":"10.1007/978-3-319-58002-9_5","day":"05","publication_status":"published","department":[{"_id":"JaMa"}],"month":"10","corr_author":"1","language":[{"iso":"eng"}],"oa_version":"None","quality_controlled":"1","citation":{"ieee":"J. Maas, “Entropic Ricci curvature for discrete spaces,” in Modern Approaches to Discrete Curvature, vol. 2184, L. Najman and P. Romon, Eds. Springer, 2017, pp. 159–174.","ama":"Maas J. Entropic Ricci curvature for discrete spaces. In: Najman L, Romon P, eds. Modern Approaches to Discrete Curvature. Vol 2184. Lecture Notes in Mathematics. Springer; 2017:159-174. doi:10.1007/978-3-319-58002-9_5","apa":"Maas, J. (2017). Entropic Ricci curvature for discrete spaces. In L. Najman & P. Romon (Eds.), Modern Approaches to Discrete Curvature (Vol. 2184, pp. 159–174). Springer. https://doi.org/10.1007/978-3-319-58002-9_5","short":"J. Maas, in:, L. Najman, P. Romon (Eds.), Modern Approaches to Discrete Curvature, Springer, 2017, pp. 159–174.","mla":"Maas, Jan. “Entropic Ricci Curvature for Discrete Spaces.” Modern Approaches to Discrete Curvature, edited by Laurent Najman and Pascal Romon, vol. 2184, Springer, 2017, pp. 159–74, doi:10.1007/978-3-319-58002-9_5.","chicago":"Maas, Jan. “Entropic Ricci Curvature for Discrete Spaces.” In Modern Approaches to Discrete Curvature, edited by Laurent Najman and Pascal Romon, 2184:159–74. Lecture Notes in Mathematics. Springer, 2017. https://doi.org/10.1007/978-3-319-58002-9_5.","ista":"Maas J. 2017.Entropic Ricci curvature for discrete spaces. In: Modern Approaches to Discrete Curvature. vol. 2184, 159–174."},"status":"public","scopus_import":"1","title":"Entropic Ricci curvature for discrete spaces","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publication_identifier":{"eisbn":["9783319580029"],"isbn":["9783319580012"]},"date_created":"2018-12-11T11:47:42Z","publication":"Modern Approaches to Discrete Curvature","abstract":[{"lang":"eng","text":"We give a short overview on a recently developed notion of Ricci curvature for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani."}],"series_title":"Lecture Notes in Mathematics","date_updated":"2026-04-16T08:59:01Z","type":"book_chapter","date_published":"2017-10-05T00:00:00Z","editor":[{"first_name":"Laurent","full_name":"Najman, Laurent","last_name":"Najman"},{"last_name":"Romon","full_name":"Romon, Pascal","first_name":"Pascal"}],"page":"159 - 174","article_processing_charge":"No","publisher":"Springer","year":"2017","_id":"649","author":[{"first_name":"Jan","orcid":"0000-0002-0845-1338","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan"}],"publist_id":"7123","intvolume":" 2184","volume":2184}]