[{"intvolume":"        21","status":"public","day":"04","type":"journal_article","publication":"Communications in Information and Systems","issue":"4","page":"481-536","publisher":"International Press","language":[{"iso":"eng"}],"month":"06","date_published":"2021-06-04T00:00:00Z","article_type":"original","date_created":"2021-09-19T08:53:19Z","department":[{"_id":"JaMa"}],"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"}],"keyword":["Markov Chain","relative entropy","time reversal","steepest descent","gradient flow"],"author":[{"full_name":"Karatzas, Ioannis","last_name":"Karatzas","first_name":"Ioannis"},{"id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338","full_name":"Maas, Jan","last_name":"Maas"},{"first_name":"Walter","full_name":"Schachermayer, Walter","last_name":"Schachermayer"}],"citation":{"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>.","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.","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.","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>","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>."},"publication_status":"published","publication_identifier":{"issn":["1526-7555"]},"_id":"10023","quality_controlled":"1","project":[{"name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"716117"},{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"oa_version":"Preprint","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","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.","arxiv":1,"article_processing_charge":"No","volume":21,"date_updated":"2021-09-20T12:51:18Z","oa":1,"external_id":{"arxiv":["2005.14177"]},"title":"Trajectorial dissipation and gradient flow for the relative entropy in Markov chains","year":"2021","doi":"10.4310/CIS.2021.v21.n4.a1","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/2005.14177","open_access":"1"}]},{"department":[{"_id":"GaTk"}],"tmp":{"name":"Creative Commons Public Domain Dedication (CC0 1.0)","short":"CC0 (1.0)","image":"/images/cc_0.png","legal_code_url":"https://creativecommons.org/publicdomain/zero/1.0/legalcode"},"has_accepted_license":"1","ddc":["530"],"file":[{"file_name":"IST-2018-111-v1+1_CODES.zip","file_size":14376,"date_created":"2018-12-12T13:05:13Z","checksum":"97992e3e8cf8544ec985a48971708726","date_updated":"2020-07-14T12:47:08Z","access_level":"open_access","content_type":"application/zip","relation":"main_file","file_id":"5641","creator":"system"}],"date_created":"2018-12-12T12:31:41Z","related_material":{"record":[{"status":"public","relation":"research_paper","id":"161"}]},"date_published":"2018-09-21T00:00:00Z","ec_funded":1,"year":"2018","month":"09","doi":"10.15479/AT:ISTA:62","title":"Supporting materials \"STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH\"","datarep_id":"111","publisher":"Institute of Science and Technology Austria","article_processing_charge":"No","file_date_updated":"2020-07-14T12:47:08Z","date_updated":"2024-02-21T13:45:39Z","oa":1,"oa_version":"Published Version","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"},{"grant_number":"P28844-B27","call_identifier":"FWF","name":"Biophysics of information processing in gene regulation","_id":"254E9036-B435-11E9-9278-68D0E5697425"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"5587","type":"research_data","day":"21","citation":{"ista":"De Martino D, Tkačik G. 2018. Supporting materials ‘STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH’, Institute of Science and Technology Austria, <a href=\"https://doi.org/10.15479/AT:ISTA:62\">10.15479/AT:ISTA:62</a>.","short":"D. De Martino, G. Tkačik, (2018).","ama":"De Martino D, Tkačik G. Supporting materials “STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.” 2018. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:62\">10.15479/AT:ISTA:62</a>","mla":"De Martino, Daniele, and Gašper Tkačik. <i>Supporting Materials “STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.”</i> Institute of Science and Technology Austria, 2018, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:62\">10.15479/AT:ISTA:62</a>.","chicago":"De Martino, Daniele, and Gašper Tkačik. “Supporting Materials ‘STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.’” Institute of Science and Technology Austria, 2018. <a href=\"https://doi.org/10.15479/AT:ISTA:62\">https://doi.org/10.15479/AT:ISTA:62</a>.","apa":"De Martino, D., &#38; Tkačik, G. (2018). Supporting materials “STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.” Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:62\">https://doi.org/10.15479/AT:ISTA:62</a>","ieee":"D. De Martino and G. Tkačik, “Supporting materials ‘STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.’” Institute of Science and Technology Austria, 2018."},"keyword":["metabolic networks","e.coli core","maximum entropy","monte carlo markov chain sampling","ellipsoidal rounding"],"status":"public","author":[{"first_name":"Daniele","last_name":"De Martino","full_name":"De Martino, Daniele","orcid":"0000-0002-5214-4706","id":"3FF5848A-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Gasper","orcid":"0000-0002-6699-1455","full_name":"Tkacik, Gasper","last_name":"Tkacik","id":"3D494DCA-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"Supporting material to the article \r\nSTATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH\r\n\r\nboundscoli.dat\r\nFlux Bounds of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium. \r\n\r\npolcoli.dat\r\nMatrix enconding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium, \r\nobtained from the soichiometric matrix by standard linear algebra  (reduced row echelon form).\r\n\r\nellis.dat\r\nApproximate Lowner-John ellipsoid rounding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium\r\nobtained with the Lovasz method.\r\n\r\npoint0.dat\r\nCenter of the approximate Lowner-John ellipsoid rounding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium\r\nobtained with the Lovasz method.\r\n\r\nlovasz.cpp  \r\nThis c++ code file receives in input the polytope of the feasible steady states of a metabolic network, \r\n(matrix and bounds), and it gives in output an approximate Lowner-John ellipsoid rounding the polytope\r\nwith the Lovasz method \r\nNB inputs are referred by defaults to the catabolic core of the E.Coli network iAF1260. \r\nFor further details we refer to  PLoS ONE 10.4 e0122670 (2015).\r\n\r\nsampleHRnew.cpp  \r\nThis c++ code file receives in input the polytope of the feasible steady states of a metabolic network, \r\n(matrix and bounds), the ellipsoid rounding the polytope, a point inside and  \r\nit gives in output a max entropy sampling at fixed average growth rate \r\nof the steady states by performing an Hit-and-Run Monte Carlo Markov chain.\r\nNB inputs are referred by defaults to the catabolic core of the E.Coli network iAF1260. \r\nFor further details we refer to  PLoS ONE 10.4 e0122670 (2015)."}]}]