[{"month":"06","date_created":"2021-09-19T08:53:19Z","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."}],"year":"2021","citation":{"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 of Boston, 2021. <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 of Boston, 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.","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 of Boston, 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 of Boston. <a href=\"https://doi.org/10.4310/CIS.2021.v21.n4.a1\">https://doi.org/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>"},"quality_controlled":"1","page":"481-536","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2005.14177"}],"type":"journal_article","publication":"Communications in Information and Systems","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"        21","publication_identifier":{"issn":["1526-7555"]},"publication_status":"published","keyword":["Markov Chain","relative entropy","time reversal","steepest descent","gradient flow"],"das_tickbox":"1","date_published":"2021-06-04T00:00:00Z","_id":"10023","arxiv":1,"publisher":"International Press of Boston","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","oa":1,"issue":"4","status":"public","external_id":{"arxiv":["2005.14177"]},"language":[{"iso":"eng"}],"department":[{"_id":"JaMa"}],"doi":"10.4310/CIS.2021.v21.n4.a1","author":[{"first_name":"Ioannis","last_name":"Karatzas","full_name":"Karatzas, Ioannis"},{"full_name":"Maas, Jan","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","orcid":"0000-0002-0845-1338"},{"last_name":"Schachermayer","full_name":"Schachermayer, Walter","first_name":"Walter"}],"day":"04","date_updated":"2026-07-06T13:38:10Z","title":"Trajectorial dissipation and gradient flow for the relative entropy in Markov chains","project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020"},{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"ec_funded":1,"oa_version":"Preprint","volume":21,"article_type":"original"},{"type":"research_data","date_created":"2018-12-12T12:31:41Z","abstract":[{"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).","lang":"eng"}],"year":"2018","citation":{"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.","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>.","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>","short":"D. De Martino, G. Tkačik, (2018).","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>","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>.","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>."},"month":"09","keyword":["metabolic networks","e.coli core","maximum entropy","monte carlo markov chain sampling","ellipsoidal rounding"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","datarep_id":"111","tmp":{"short":"CC0 (1.0)","legal_code_url":"https://creativecommons.org/publicdomain/zero/1.0/legalcode","image":"/images/cc_0.png","name":"Creative Commons Public Domain Dedication (CC0 1.0)"},"file":[{"file_id":"5641","file_name":"IST-2018-111-v1+1_CODES.zip","relation":"main_file","date_created":"2018-12-12T13:05:13Z","date_updated":"2020-07-14T12:47:08Z","file_size":14376,"creator":"system","access_level":"open_access","content_type":"application/zip","checksum":"97992e3e8cf8544ec985a48971708726"}],"status":"public","publisher":"Institute of Science and Technology Austria","article_processing_charge":"No","oa":1,"date_published":"2018-09-21T00:00:00Z","_id":"5587","oa_version":"Published Version","file_date_updated":"2020-07-14T12:47:08Z","title":"Supporting materials \"STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH\"","date_updated":"2025-04-15T06:50:08Z","license":"https://creativecommons.org/publicdomain/zero/1.0/","ec_funded":1,"project":[{"call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"},{"_id":"254E9036-B435-11E9-9278-68D0E5697425","grant_number":"P28844-B27","name":"Biophysics of information processing in gene regulation","call_identifier":"FWF"}],"doi":"10.15479/AT:ISTA:62","has_accepted_license":"1","author":[{"id":"3FF5848A-F248-11E8-B48F-1D18A9856A87","last_name":"De Martino","full_name":"De Martino, Daniele","orcid":"0000-0002-5214-4706","first_name":"Daniele"},{"id":"3D494DCA-F248-11E8-B48F-1D18A9856A87","last_name":"Tkacik","full_name":"Tkacik, Gasper","orcid":"0000-0002-6699-1455","first_name":"Gasper"}],"ddc":["530"],"day":"21","related_material":{"record":[{"relation":"research_paper","id":"161","status":"public"}]},"department":[{"_id":"GaTk"}]},{"das_tickbox":"1","date_published":"2014-01-01T00:00:00Z","_id":"22575","publisher":"Elsevier","article_processing_charge":"No","status":"public","language":[{"iso":"eng"}],"doi":"10.1016/j.advwatres.2013.11.006","scopus_import":"1","OA_type":"closed access","author":[{"full_name":"Paschalis, Athanasios","last_name":"Paschalis","first_name":"Athanasios"},{"last_name":"Molnar","full_name":"Molnar, Peter","first_name":"Peter"},{"first_name":"Simone","id":"cf8e546b-a9b0-11f0-a43b-aa89ed1b56d6","full_name":"Fatichi, Simone","last_name":"Fatichi"},{"last_name":"Burlando","full_name":"Burlando, Paolo","first_name":"Paolo"}],"day":"01","date_updated":"2026-08-10T11:55:17Z","title":"On temporal stochastic modeling of precipitation, nesting models across scales","volume":63,"oa_version":"None","article_type":"original","extern":"1","month":"01","date_created":"2026-07-27T12:30:25Z","abstract":[{"lang":"eng","text":"We analyze the performance of composite stochastic models of temporal precipitation which can satisfactorily reproduce precipitation properties across a wide range of temporal scales. The rationale is that a combination of stochastic precipitation models which are most appropriate for specific limited temporal scales leads to better overall performance across a wider range of scales than single models alone. We investigate different model combinations. For the coarse (daily) scale these are models based on Alternating renewal processes, Markov chains, and Poisson cluster models, which are then combined with a microcanonical Multiplicative Random Cascade model to disaggregate precipitation to finer (minute) scales. The composite models were tested on data at four sites in different climates. The results show that model combinations improve the performance in key statistics such as probability distributions of precipitation depth, autocorrelation structure, intermittency, reproduction of extremes, compared to single models. At the same time they remain reasonably parsimonious. No model combination was found to outperform the others at all sites and for all statistics, however we provide insight on the capabilities of specific model combinations. The results for the four different climates are similar, which suggests a degree of generality and wider applicability of the approach."}],"quality_controlled":"1","citation":{"ieee":"A. Paschalis, P. Molnar, S. Fatichi, and P. Burlando, “On temporal stochastic modeling of precipitation, nesting models across scales,” <i>Advances in Water Resources</i>, vol. 63. Elsevier, pp. 152–166, 2014.","chicago":"Paschalis, Athanasios, Peter Molnar, Simone Fatichi, and Paolo Burlando. “On Temporal Stochastic Modeling of Precipitation, Nesting Models across Scales.” <i>Advances in Water Resources</i>. Elsevier, 2014. <a href=\"https://doi.org/10.1016/j.advwatres.2013.11.006\">https://doi.org/10.1016/j.advwatres.2013.11.006</a>.","ama":"Paschalis A, Molnar P, Fatichi S, Burlando P. On temporal stochastic modeling of precipitation, nesting models across scales. <i>Advances in Water Resources</i>. 2014;63:152-166. doi:<a href=\"https://doi.org/10.1016/j.advwatres.2013.11.006\">10.1016/j.advwatres.2013.11.006</a>","mla":"Paschalis, Athanasios, et al. “On Temporal Stochastic Modeling of Precipitation, Nesting Models across Scales.” <i>Advances in Water Resources</i>, vol. 63, Elsevier, 2014, pp. 152–66, doi:<a href=\"https://doi.org/10.1016/j.advwatres.2013.11.006\">10.1016/j.advwatres.2013.11.006</a>.","apa":"Paschalis, A., Molnar, P., Fatichi, S., &#38; Burlando, P. (2014). On temporal stochastic modeling of precipitation, nesting models across scales. <i>Advances in Water Resources</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.advwatres.2013.11.006\">https://doi.org/10.1016/j.advwatres.2013.11.006</a>","short":"A. Paschalis, P. Molnar, S. Fatichi, P. Burlando, Advances in Water Resources 63 (2014) 152–166.","ista":"Paschalis A, Molnar P, Fatichi S, Burlando P. 2014. On temporal stochastic modeling of precipitation, nesting models across scales. Advances in Water Resources. 63, 152–166."},"year":"2014","page":"152-166","publication":"Advances in Water Resources","type":"journal_article","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","intvolume":"        63","publication_identifier":{"issn":["0309-1708"]},"publication_status":"published","keyword":["Stochastic rainfall models","Poisson cluster model","Multiplicative Random Cascade","Markov chain","Alternating renewal process"]}]