--- OA_place: repository OA_type: green _id: '20050' abstract: - lang: eng 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. 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. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Kseniia full_name: Khudiakova, Kseniia id: 4E6DC800-AE37-11E9-AC72-31CAE5697425 last_name: Khudiakova orcid: 0000-0002-6246-1465 - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Francesco full_name: Pedrotti, Francesco id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c last_name: Pedrotti citation: 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 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 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. 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. 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. short: K. Khudiakova, J. Maas, F. Pedrotti, The Annals of Applied Probability 35 (2025) 1913–1940. corr_author: '1' date_created: 2025-07-21T08:13:54Z date_published: 2025-06-01T00:00:00Z date_updated: 2025-09-30T14:12:48Z day: '01' department: - _id: JaMa doi: 10.1214/25-aap2162 external_id: arxiv: - '2402.04151' isi: - '001523520000012' intvolume: ' 35' isi: 1 issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2402.04151 month: '06' oa: 1 oa_version: Preprint page: 1913-1940 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 publication: The Annals of Applied Probability publication_identifier: issn: - 1050-5164 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' related_material: record: - id: '17352' relation: earlier_version status: public scopus_import: '1' status: public title: L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 35 year: '2025' ... --- OA_place: repository OA_type: green _id: '20569' abstract: - lang: eng 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.' 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." article_number: '2502.15665' article_processing_charge: No arxiv: 1 author: - first_name: Giovanni full_name: Brigati, Giovanni id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1 last_name: Brigati - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Filippo full_name: Quattrocchi, Filippo id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308 last_name: Quattrocchi orcid: 0009-0000-9773-1931 citation: 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' 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' 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.' 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.' 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.' short: G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.). corr_author: '1' date_created: 2025-10-28T13:12:08Z date_published: 2025-08-10T00:00:00Z date_updated: 2026-06-24T22:30:42Z day: '10' department: - _id: GradSch - _id: JaMa doi: 10.48550/arXiv.2502.15665 ec_funded: 1 external_id: arxiv: - '2502.15665' keyword: - optimal transport - kinetic theory - second-order discrepancy - Vlasov equation - Wasserstein splines. language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2502.15665 month: '08' oa: 1 oa_version: Preprint project: - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c call_identifier: H2020 grant_number: '101034413' name: 'IST-BRIDGE: International postdoctoral program' publication: arXiv publication_status: draft related_material: record: - id: '20563' relation: dissertation_contains status: public status: public title: 'Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures' type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2025' ... --- _id: '17143' 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." 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 article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Lorenzo full_name: Dello Schiavo, Lorenzo id: ECEBF480-9E4F-11EA-B557-B0823DDC885E last_name: Dello Schiavo orcid: 0000-0002-9881-6870 - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Francesco full_name: Pedrotti, Francesco id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c last_name: Pedrotti citation: 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 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 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. 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. 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. date_created: 2024-06-16T22:01:06Z date_published: 2024-06-01T00:00:00Z date_updated: 2026-04-07T13:00:02Z day: '01' department: - _id: JaMa doi: 10.1090/tran/9156 ec_funded: 1 external_id: arxiv: - '2304.05239' isi: - '001203273300001' intvolume: ' 377' isi: 1 issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2304.05239 month: '06' oa: 1 oa_version: Preprint page: 3779-3804 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: 34dbf174-11ca-11ed-8bc3-afe9d43d4b9c grant_number: E208 name: Configuration Spaces over Non-Smooth Spaces publication: Transactions of the American Mathematical Society publication_identifier: eissn: - 1088-6850 issn: - 0002-9947 publication_status: published publisher: American Mathematical Society quality_controlled: '1' related_material: record: - id: '17336' relation: dissertation_contains status: public scopus_import: '1' status: public title: Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 377 year: '2024' ... --- OA_place: publisher OA_type: gold _id: '18897' 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.' 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" alternative_title: - TMLR article_processing_charge: No arxiv: 1 author: - first_name: Francesco full_name: Pedrotti, Francesco id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c last_name: Pedrotti - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Marco full_name: Mondelli, Marco id: 27EB676C-8706-11E9-9510-7717E6697425 last_name: Mondelli orcid: 0000-0002-3242-7020 citation: 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. 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. 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. ista: Pedrotti F, Maas J, Mondelli M. 2024. Improved convergence of score-based diffusion models via prediction-correction. Transactions on Machine Learning Research. , TMLR, . mla: Pedrotti, Francesco, et al. “Improved Convergence of Score-Based Diffusion Models via Prediction-Correction.” Transactions on Machine Learning Research, 2024. short: F. Pedrotti, J. Maas, M. Mondelli, in:, Transactions on Machine Learning Research, 2024. corr_author: '1' date_created: 2025-01-27T12:18:05Z date_published: 2024-06-01T00:00:00Z date_updated: 2025-04-15T08:31:35Z day: '01' ddc: - '000' department: - _id: JaMa - _id: MaMo external_id: arxiv: - '2305.14164' file: - access_level: open_access checksum: 76a1fd5afd8ee6f7ae0e5912d7dbf6b4 content_type: application/pdf creator: dernst date_created: 2025-01-27T12:19:44Z date_updated: 2025-01-27T12:19:44Z file_id: '18898' file_name: 2024_TMLR_Pedrotti.pdf file_size: 780315 relation: main_file success: 1 file_date_updated: 2025-01-27T12:19:44Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Published Version 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 publication: Transactions on Machine Learning Research publication_identifier: issn: - 2835-8856 publication_status: published quality_controlled: '1' related_material: record: - id: '17350' relation: earlier_version status: public scopus_import: '1' status: public title: Improved convergence of score-based diffusion models via prediction-correction tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- _id: '18899' 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." alternative_title: - Bolyai Society Mathematical Studies article_processing_charge: No 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' 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.' 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. date_created: 2025-01-27T12:26:03Z date_published: 2024-09-19T00:00:00Z date_updated: 2025-02-17T12:22:18Z day: '19' department: - _id: JaMa doi: 10.1007/978-3-031-50466-2 editor: - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Simone Anna Elvira full_name: Rademacher, Simone Anna Elvira id: 856966FE-A408-11E9-977E-802DE6697425 last_name: Rademacher orcid: 0000-0001-5059-4466 - first_name: Tamás full_name: Titkos, Tamás last_name: Titkos - first_name: Daniel full_name: Virosztek, Daniel id: 48DB45DA-F248-11E8-B48F-1D18A9856A87 last_name: Virosztek orcid: 0000-0003-1109-5511 intvolume: ' 29' language: - iso: eng month: '09' oa_version: None place: Cham publication_identifier: eisbn: - '9783031504662' eissn: - 2947-9460 isbn: - '9783031504655' issn: - 1217-4696 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' series_title: BSMS status: public title: Optimal Transport on Quantum Structures type: book_editor user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 29 year: '2024' ... --- _id: '17282' abstract: - lang: eng 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. 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. article_number: '153' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Morris full_name: Brooks, Morris id: B7ECF9FC-AA38-11E9-AC9A-0930E6697425 last_name: Brooks orcid: 0000-0002-6249-0928 - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: 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 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. 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. ista: Brooks M, Maas J. 2024. Characterisation of gradient flows for a given functional. Calculus of Variations and Partial Differential Equations. 63(6), 153. 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). corr_author: '1' date_created: 2024-07-21T22:01:01Z date_published: 2024-07-01T00:00:00Z date_updated: 2025-09-08T08:24:51Z day: '01' ddc: - '510' department: - _id: JaMa doi: 10.1007/s00526-024-02755-z ec_funded: 1 external_id: arxiv: - '2209.11149' isi: - '001258097800003' pmid: - '38947856' file: - access_level: open_access checksum: a0cf0e0ba2157aabb287cb597be17dac content_type: application/pdf creator: dernst date_created: 2024-07-22T07:05:32Z date_updated: 2024-07-22T07:05:32Z file_id: '17289' file_name: 2024_CalculusVariations_Brooks.pdf file_size: 416622 relation: main_file success: 1 file_date_updated: 2024-07-22T07:05:32Z has_accepted_license: '1' intvolume: ' 63' isi: 1 issue: '6' language: - iso: eng month: '07' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: Calculus of Variations and Partial Differential Equations publication_identifier: eissn: - 1432-0835 issn: - 0944-2669 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Characterisation of gradient flows for a given functional tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 63 year: '2024' ... --- OA_place: repository _id: '17350' 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." article_processing_charge: No arxiv: 1 author: - first_name: Francesco full_name: Pedrotti, Francesco id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c last_name: Pedrotti - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Marco full_name: Mondelli, Marco id: 27EB676C-8706-11E9-9510-7717E6697425 last_name: Mondelli orcid: 0000-0002-3242-7020 citation: ama: Pedrotti F, Maas J, Mondelli M. Improved convergence of score-based diffusion models via prediction-correction. arXiv. doi:10.48550/arXiv.2305.14164 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 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. ieee: F. Pedrotti, J. Maas, and M. Mondelli, “Improved convergence of score-based diffusion models via prediction-correction,” arXiv. . 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. short: F. Pedrotti, J. Maas, M. Mondelli, ArXiv (n.d.). corr_author: '1' date_created: 2024-07-31T07:56:40Z date_published: 2024-06-06T00:00:00Z date_updated: 2026-04-07T13:00:02Z day: '06' department: - _id: JaMa - _id: MaMo doi: 10.48550/arXiv.2305.14164 external_id: arxiv: - '2305.14164' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2305.14164 month: '06' oa: 1 oa_version: Preprint 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 publication: arXiv publication_status: draft related_material: record: - id: '18897' relation: later_version status: public - id: '17336' relation: dissertation_contains status: public status: public title: Improved convergence of score-based diffusion models via prediction-correction type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: repository _id: '17352' abstract: - lang: eng 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." article_number: '2402.04151' article_processing_charge: No arxiv: 1 author: - first_name: Kseniia full_name: Khudiakova, Kseniia id: 4E6DC800-AE37-11E9-AC72-31CAE5697425 last_name: Khudiakova orcid: 0000-0002-6246-1465 - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Francesco full_name: Pedrotti, Francesco id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c last_name: Pedrotti citation: 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 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 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. 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. . 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. 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. short: K. Khudiakova, J. Maas, F. Pedrotti, ArXiv (n.d.). corr_author: '1' date_created: 2024-07-31T08:07:40Z date_published: 2024-02-07T00:00:00Z date_updated: 2026-04-07T13:00:02Z day: '07' department: - _id: JaMa doi: 10.48550/arXiv.2402.04151 external_id: arxiv: - '2402.04151' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2402.04151 month: '02' oa: 1 oa_version: Preprint 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 publication: arXiv publication_status: draft related_material: record: - id: '20050' relation: later_version status: public - id: '17336' relation: dissertation_contains status: public status: public title: L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- _id: '12959' 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." 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). article_number: '143' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Peter full_name: Gladbach, Peter last_name: Gladbach - first_name: Eva full_name: Kopfer, Eva last_name: Kopfer - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale citation: 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 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 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. 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. 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. short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and Partial Differential Equations 62 (2023). corr_author: '1' date_created: 2023-05-14T22:01:00Z date_published: 2023-04-28T00:00:00Z date_updated: 2025-05-15T10:54:12Z day: '28' ddc: - '510' department: - _id: JaMa doi: 10.1007/s00526-023-02472-z ec_funded: 1 external_id: arxiv: - '2110.15321' isi: - '000980588900001' pmid: - '37131846' file: - access_level: open_access checksum: 359bee38d94b7e0aa73925063cb8884d content_type: application/pdf creator: dernst date_created: 2023-10-04T11:34:10Z date_updated: 2023-10-04T11:34:10Z file_id: '14393' file_name: 2023_CalculusEquations_Gladbach.pdf file_size: 1240995 relation: main_file success: 1 file_date_updated: 2023-10-04T11:34:10Z has_accepted_license: '1' intvolume: ' 62' isi: 1 issue: '5' language: - iso: eng month: '04' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: 260788DE-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: W1245 name: Dissipation and dispersion in nonlinear partial differential equations publication: Calculus of Variations and Partial Differential Equations publication_identifier: eissn: - 1432-0835 issn: - 0944-2669 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Homogenisation of dynamical optimal transport on periodic graphs tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 62 year: '2023' ... --- _id: '11700' 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. 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." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Matthias full_name: Erbar, Matthias last_name: Erbar - first_name: Dominik L full_name: Forkert, Dominik L id: 35C79D68-F248-11E8-B48F-1D18A9856A87 last_name: Forkert - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Delio full_name: Mugnolo, Delio last_name: Mugnolo citation: 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 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 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. 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. corr_author: '1' date_created: 2022-07-31T22:01:46Z date_published: 2022-10-01T00:00:00Z date_updated: 2025-04-14T07:27:47Z day: '01' department: - _id: JaMa doi: 10.3934/nhm.2022023 ec_funded: 1 external_id: arxiv: - '2105.05677' isi: - '000812422100001' intvolume: ' 17' isi: 1 issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2105.05677 month: '10' oa: 1 oa_version: Preprint page: 687-717 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: Networks and Heterogeneous Media publication_identifier: eissn: - 1556-181X issn: - 1556-1801 publication_status: published publisher: American Institute of Mathematical Sciences quality_controlled: '1' scopus_import: '1' status: public title: Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 17 year: '2022' ... --- _id: '11739' abstract: - lang: eng 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. 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. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Dominik L full_name: Forkert, Dominik L id: 35C79D68-F248-11E8-B48F-1D18A9856A87 last_name: Forkert - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale citation: 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. 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. 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. 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. corr_author: '1' date_created: 2022-08-07T22:01:59Z date_published: 2022-07-18T00:00:00Z date_updated: 2025-04-15T08:31:31Z day: '18' department: - _id: JaMa doi: 10.1137/21M1410968 ec_funded: 1 external_id: arxiv: - '2008.10962' isi: - '000889274600001' intvolume: ' 54' isi: 1 issue: '4' keyword: - Fokker--Planck equation - gradient flow - evolutionary $\Gamma$-convergence language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.48550/arXiv.2008.10962' month: '07' oa: 1 oa_version: Preprint page: 4297-4333 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: 260788DE-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: W1245 name: Dissipation and dispersion in nonlinear partial differential equations publication: SIAM Journal on Mathematical Analysis publication_identifier: eissn: - 1095-7154 issn: - 0036-1410 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' related_material: record: - id: '10022' relation: earlier_version status: public scopus_import: '1' status: public title: Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 54 year: '2022' ... --- _id: '10023' abstract: - lang: eng 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. 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. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Ioannis full_name: Karatzas, Ioannis last_name: Karatzas - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Walter full_name: Schachermayer, Walter last_name: Schachermayer citation: 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 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 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. 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. short: I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and Systems 21 (2021) 481–536. date_created: 2021-09-19T08:53:19Z date_published: 2021-06-04T00:00:00Z date_updated: 2025-04-14T07:27:45Z day: '04' department: - _id: JaMa doi: 10.4310/CIS.2021.v21.n4.a1 ec_funded: 1 external_id: arxiv: - '2005.14177' intvolume: ' 21' issue: '4' keyword: - Markov Chain - relative entropy - time reversal - steepest descent - gradient flow language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2005.14177 month: '06' oa: 1 oa_version: Preprint page: 481-536 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: Communications in Information and Systems publication_identifier: issn: - 1526-7555 publication_status: published publisher: International Press quality_controlled: '1' status: public title: Trajectorial dissipation and gradient flow for the relative entropy in Markov chains type: journal_article user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 21 year: '2021' ... --- _id: '6358' 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. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Eric A. full_name: Carlen, Eric A. last_name: Carlen - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: 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 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 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. 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. short: E.A. Carlen, J. Maas, Journal of Statistical Physics 178 (2020) 319–378. corr_author: '1' date_created: 2019-04-30T07:34:18Z date_published: 2020-01-01T00:00:00Z date_updated: 2025-06-12T07:27:20Z day: '01' ddc: - '500' department: - _id: JaMa doi: 10.1007/s10955-019-02434-w ec_funded: 1 external_id: arxiv: - '1811.04572' isi: - '000498933300001' pmid: - '33223567' file: - access_level: open_access checksum: 7b04befbdc0d4982c0ee945d25d19872 content_type: application/pdf creator: dernst date_created: 2019-12-23T12:03:09Z date_updated: 2020-07-14T12:47:28Z file_id: '7209' file_name: 2019_JourStatistPhysics_Carlen.pdf file_size: 905538 relation: main_file file_date_updated: 2020-07-14T12:47:28Z has_accepted_license: '1' intvolume: ' 178' isi: 1 issue: '2' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 319-378 pmid: 1 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems publication: Journal of Statistical Physics publication_identifier: eissn: - 1572-9613 issn: - 0022-4715 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: link: - relation: erratum url: https://doi.org/10.1007/s10955-020-02671-4 scopus_import: '1' status: public title: Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 178 year: '2020' ... --- _id: '10022' abstract: - lang: eng 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. 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. article_number: '2008.10962' article_processing_charge: No arxiv: 1 author: - first_name: Dominik L full_name: Forkert, Dominik L id: 35C79D68-F248-11E8-B48F-1D18A9856A87 last_name: Forkert - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale citation: 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 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 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. 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. short: D.L. Forkert, J. Maas, L. Portinale, ArXiv (n.d.). corr_author: '1' date_created: 2021-09-17T10:57:27Z date_published: 2020-08-25T00:00:00Z date_updated: 2026-04-08T07:00:03Z day: '25' department: - _id: JaMa doi: 10.48550/arXiv.2008.10962 ec_funded: 1 external_id: arxiv: - '2008.10962' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2008.10962 month: '08' oa: 1 oa_version: Preprint project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: arXiv publication_status: draft related_material: record: - id: '11739' relation: later_version status: public - id: '10030' relation: dissertation_contains status: public status: public title: Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2020' ... --- _id: '7573' 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. 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. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Peter full_name: Gladbach, Peter last_name: Gladbach - first_name: Eva full_name: Kopfer, Eva last_name: Kopfer - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale citation: 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 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 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. 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. 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. 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. short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Journal de Mathematiques Pures et Appliquees 139 (2020) 204–234. date_created: 2020-03-08T23:00:47Z date_published: 2020-07-01T00:00:00Z date_updated: 2026-04-08T07:00:03Z day: '01' department: - _id: JaMa doi: 10.1016/j.matpur.2020.02.008 ec_funded: 1 external_id: arxiv: - '1905.05757' isi: - '000539439400008' intvolume: ' 139' isi: 1 issue: '7' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1905.05757 month: '07' oa: 1 oa_version: Preprint page: 204-234 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems - _id: 260788DE-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: W1245 name: Dissipation and dispersion in nonlinear partial differential equations publication: Journal de Mathematiques Pures et Appliquees publication_identifier: issn: - 0021-7824 publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '10030' relation: dissertation_contains status: public scopus_import: '1' status: public title: Homogenisation of one-dimensional discrete optimal transport type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 139 year: '2020' ... --- _id: '8758' abstract: - lang: eng 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. 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). article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Alexander full_name: Mielke, Alexander last_name: Mielke citation: 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 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 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. 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. 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. 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. short: J. Maas, A. Mielke, Journal of Statistical Physics 181 (2020) 2257–2303. corr_author: '1' date_created: 2020-11-15T23:01:18Z date_published: 2020-12-01T00:00:00Z date_updated: 2025-06-12T07:01:39Z day: '01' ddc: - '510' department: - _id: JaMa doi: 10.1007/s10955-020-02663-4 ec_funded: 1 external_id: arxiv: - '2004.02831' isi: - '000587107200002' pmid: - '33268907' file: - access_level: open_access checksum: bc2b63a90197b97cbc73eccada4639f5 content_type: application/pdf creator: dernst date_created: 2021-02-04T10:29:11Z date_updated: 2021-02-04T10:29:11Z file_id: '9087' file_name: 2020_JourStatPhysics_Maas.pdf file_size: 753596 relation: main_file success: 1 file_date_updated: 2021-02-04T10:29:11Z has_accepted_license: '1' intvolume: ' 181' isi: 1 issue: '6' language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 2257-2303 pmid: 1 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems publication: Journal of Statistical Physics publication_identifier: eissn: - 1572-9613 issn: - 0022-4715 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Modeling of chemical reaction systems with detailed balance using gradient structures tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 181 year: '2020' ... --- _id: '71' 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 \U0001D54E2, 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 \U0001D54E2, 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." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Peter full_name: Gladbach, Peter last_name: Gladbach - first_name: Eva full_name: Kopfer, Eva last_name: Kopfer - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: 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 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 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. 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. ista: Gladbach P, Kopfer E, Maas J. 2020. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis. 52(3), 2759–2802. 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. date_created: 2018-12-11T11:44:28Z date_published: 2020-10-01T00:00:00Z date_updated: 2025-07-10T11:54:14Z day: '01' department: - _id: JaMa doi: 10.1137/19M1243440 external_id: arxiv: - '1809.01092' isi: - '000546975100017' intvolume: ' 52' isi: 1 issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1809.01092 month: '10' oa: 1 oa_version: Preprint page: 2759-2802 publication: SIAM Journal on Mathematical Analysis publication_identifier: eissn: - 1095-7154 issn: - 0036-1410 publication_status: published publisher: Society for Industrial and Applied Mathematics publist_id: '7983' quality_controlled: '1' scopus_import: '1' status: public title: Scaling limits of discrete optimal transport type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 52 year: '2020' ... --- _id: '73' 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. article_number: '19' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Matthias full_name: Erbar, Matthias last_name: Erbar - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Melchior full_name: Wirth, Melchior last_name: Wirth citation: 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 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 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. 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. 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. 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). date_created: 2018-12-11T11:44:29Z date_published: 2019-02-01T00:00:00Z date_updated: 2026-04-16T09:51:42Z day: '01' ddc: - '510' department: - _id: JaMa doi: 10.1007/s00526-018-1456-1 ec_funded: 1 external_id: arxiv: - '1805.06040' isi: - '000452849400001' file: - access_level: open_access checksum: ba05ac2d69de4c58d2cd338b63512798 content_type: application/pdf creator: dernst date_created: 2019-01-28T15:37:11Z date_updated: 2020-07-14T12:47:55Z file_id: '5895' file_name: 2018_Calculus_Erbar.pdf file_size: 645565 relation: main_file file_date_updated: 2020-07-14T12:47:55Z has_accepted_license: '1' intvolume: ' 58' isi: 1 issue: '1' language: - iso: eng month: '02' oa: 1 oa_version: Published Version project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Calculus of Variations and Partial Differential Equations publication_identifier: issn: - 0944-2669 publication_status: published publisher: Springer quality_controlled: '1' scopus_import: '1' status: public title: On the geometry of geodesics in discrete optimal transport tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd volume: 58 year: '2019' ... --- _id: '956' 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. article_processing_charge: No arxiv: 1 author: - first_name: Eric full_name: Carlen, Eric last_name: Carlen - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: 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 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 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. 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. 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. 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. short: E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869. date_created: 2018-12-11T11:49:24Z date_published: 2017-09-01T00:00:00Z date_updated: 2025-06-04T08:14:53Z day: '01' department: - _id: JaMa doi: 10.1016/j.jfa.2017.05.003 external_id: arxiv: - '1609.01254' isi: - '000406082300005' intvolume: ' 273' isi: 1 issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1609.01254 month: '09' oa: 1 oa_version: Submitted Version page: 1810 - 1869 publication: Journal of Functional Analysis publication_identifier: issn: - 0022-1236 publication_status: published publisher: Academic Press publist_id: '6452' quality_controlled: '1' scopus_import: '1' status: public title: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 273 year: '2017' ... --- _id: '649' 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. article_processing_charge: No author: - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: 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 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. 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. ista: 'Maas J. 2017.Entropic Ricci curvature for discrete spaces. In: Modern Approaches to Discrete Curvature. vol. 2184, 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. short: J. Maas, in:, L. Najman, P. Romon (Eds.), Modern Approaches to Discrete Curvature, Springer, 2017, pp. 159–174. corr_author: '1' date_created: 2018-12-11T11:47:42Z date_published: 2017-10-05T00:00:00Z date_updated: 2026-04-16T08:59:01Z day: '05' department: - _id: JaMa doi: 10.1007/978-3-319-58002-9_5 editor: - first_name: Laurent full_name: Najman, Laurent last_name: Najman - first_name: Pascal full_name: Romon, Pascal last_name: Romon intvolume: ' 2184' language: - iso: eng month: '10' oa_version: None page: 159 - 174 publication: Modern Approaches to Discrete Curvature publication_identifier: eisbn: - '9783319580029' isbn: - '9783319580012' publication_status: published publisher: Springer publist_id: '7123' quality_controlled: '1' scopus_import: '1' series_title: Lecture Notes in Mathematics status: public title: Entropic Ricci curvature for discrete spaces type: book_chapter user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd volume: 2184 year: '2017' ...