[{"main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2008.10962","open_access":"1"}],"publication":"SIAM Journal on Mathematical Analysis","title":"Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions","external_id":{"isi":["000889274600001"],"arxiv":["2008.10962"]},"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"10022"}]},"citation":{"ama":"Forkert DL, Maas J, Portinale L. Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(4):4297-4333. doi:<a href=\"https://doi.org/10.1137/21M1410968\">10.1137/21M1410968</a>","apa":"Forkert, D. L., Maas, J., &#38; Portinale, L. (2022). Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/21M1410968\">https://doi.org/10.1137/21M1410968</a>","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.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4, Society for Industrial and Applied Mathematics, 2022, pp. 4297–333, doi:<a href=\"https://doi.org/10.1137/21M1410968\">10.1137/21M1410968</a>.","short":"D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333.","ieee":"D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4. Society for Industrial and Applied Mathematics, pp. 4297–4333, 2022.","chicago":"Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2022. <a href=\"https://doi.org/10.1137/21M1410968\">https://doi.org/10.1137/21M1410968</a>."},"arxiv":1,"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","name":"Dissipation and Dispersion in Nonlinear Partial Differential Equations","call_identifier":"FWF"}],"scopus_import":"1","issue":"4","day":"18","page":"4297-4333","publication_status":"published","oa_version":"Preprint","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."}],"ec_funded":1,"date_updated":"2023-08-03T12:37:21Z","status":"public","month":"07","publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"quality_controlled":"1","isi":1,"article_type":"original","volume":54,"date_created":"2022-08-07T22:01:59Z","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.","department":[{"_id":"JaMa"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","intvolume":"        54","keyword":["Fokker--Planck equation","gradient flow","evolutionary $\\Gamma$-convergence"],"author":[{"id":"35C79D68-F248-11E8-B48F-1D18A9856A87","last_name":"Forkert","full_name":"Forkert, Dominik L","first_name":"Dominik L"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","first_name":"Jan"},{"full_name":"Portinale, Lorenzo","first_name":"Lorenzo","id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","last_name":"Portinale"}],"_id":"11739","oa":1,"date_published":"2022-07-18T00:00:00Z","doi":"10.1137/21M1410968","language":[{"iso":"eng"}],"year":"2022","publisher":"Society for Industrial and Applied Mathematics","type":"journal_article"},{"status":"public","type":"preprint","language":[{"iso":"eng"}],"doi":"10.48550/arXiv.2109.04233","date_updated":"2023-05-03T10:34:38Z","year":"2021","oa_version":"Preprint","date_published":"2021-09-09T00:00:00Z","publication_status":"submitted","day":"09","ec_funded":1,"abstract":[{"text":"We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.","lang":"eng"}],"author":[{"full_name":"Hensel, Sebastian","first_name":"Sebastian","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","last_name":"Hensel","orcid":"0000-0001-7252-8072"},{"full_name":"Laux, Tim","first_name":"Tim","last_name":"Laux"}],"oa":1,"_id":"10011","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","keyword":["Mean curvature flow","gradient flows","varifolds","weak solutions","weak-strong uniqueness","calibrated geometry","gradient-flow calibrations"],"date_created":"2021-09-13T12:17:10Z","article_number":"2109.04233","project":[{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020","grant_number":"948819"}],"department":[{"_id":"JuFi"}],"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. The content of this paper was developed and parts of it were written during a visit of the first author to the Hausdorff Center of Mathematics (HCM), University of Bonn. The hospitality and the support of HCM are gratefully acknowledged.","arxiv":1,"citation":{"chicago":"Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2109.04233\">https://doi.org/10.48550/arXiv.2109.04233</a>.","ieee":"S. Hensel and T. Laux, “A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness,” <i>arXiv</i>. .","mla":"Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” <i>ArXiv</i>, 2109.04233, doi:<a href=\"https://doi.org/10.48550/arXiv.2109.04233\">10.48550/arXiv.2109.04233</a>.","short":"S. Hensel, T. Laux, ArXiv (n.d.).","ista":"Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. arXiv, 2109.04233.","apa":"Hensel, S., &#38; Laux, T. (n.d.). A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2109.04233\">https://doi.org/10.48550/arXiv.2109.04233</a>","ama":"Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2109.04233\">10.48550/arXiv.2109.04233</a>"},"publication":"arXiv","external_id":{"arxiv":["2109.04233"]},"title":"A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness","month":"09","main_file_link":[{"url":"https://arxiv.org/abs/2109.04233","open_access":"1"}]},{"external_id":{"arxiv":["2005.14177"]},"title":"Trajectorial dissipation and gradient flow for the relative entropy in Markov chains","publication":"Communications in Information and Systems","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2005.14177"}],"issue":"4","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"arxiv":1,"citation":{"ama":"Karatzas I, Maas J, Schachermayer W. Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. <i>Communications in Information and Systems</i>. 2021;21(4):481-536. doi:<a href=\"https://doi.org/10.4310/CIS.2021.v21.n4.a1\">10.4310/CIS.2021.v21.n4.a1</a>","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.","short":"I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and Systems 21 (2021) 481–536.","mla":"Karatzas, Ioannis, et al. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” <i>Communications in Information and Systems</i>, vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:<a href=\"https://doi.org/10.4310/CIS.2021.v21.n4.a1\">10.4310/CIS.2021.v21.n4.a1</a>.","apa":"Karatzas, I., Maas, J., &#38; Schachermayer, W. (2021). Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. <i>Communications in Information and Systems</i>. International Press. <a href=\"https://doi.org/10.4310/CIS.2021.v21.n4.a1\">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a>","ieee":"I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and gradient flow for the relative entropy in Markov chains,” <i>Communications in Information and Systems</i>, vol. 21, no. 4. International Press, pp. 481–536, 2021.","chicago":"Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” <i>Communications in Information and Systems</i>. International Press, 2021. <a href=\"https://doi.org/10.4310/CIS.2021.v21.n4.a1\">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a>."},"ec_funded":1,"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_status":"published","page":"481-536","oa_version":"Preprint","day":"04","status":"public","date_updated":"2021-09-20T12:51:18Z","article_type":"original","volume":21,"quality_controlled":"1","publication_identifier":{"issn":["1526-7555"]},"month":"06","keyword":["Markov Chain","relative entropy","time reversal","steepest descent","gradient flow"],"intvolume":"        21","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","article_processing_charge":"No","department":[{"_id":"JaMa"}],"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.","date_created":"2021-09-19T08:53:19Z","date_published":"2021-06-04T00:00:00Z","oa":1,"_id":"10023","author":[{"full_name":"Karatzas, Ioannis","first_name":"Ioannis","last_name":"Karatzas"},{"first_name":"Jan","full_name":"Maas, Jan","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas"},{"first_name":"Walter","full_name":"Schachermayer, Walter","last_name":"Schachermayer"}],"type":"journal_article","publisher":"International Press","year":"2021","language":[{"iso":"eng"}],"doi":"10.4310/CIS.2021.v21.n4.a1"}]