[{"ddc":["510"],"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)"},"isi":1,"article_number":"9","title":"Ergodic decompositions of Dirichlet forms under order isomorphisms","external_id":{"isi":["000906214600004"]},"doi":"10.1007/s00028-022-00859-7","year":"2023","ec_funded":1,"_id":"12104","publication_identifier":{"issn":["1424-3199"],"eissn":["1424-3202"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Research supported by the Austrian Science Fund (FWF) grant F65 at the Institute of Science and Technology Austria and by the European Research Council (ERC) (Grant agreement No. 716117 awarded to Prof. Dr. Jan Maas). L.D.S. gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) through the ESPRIT Programme (Grant No. 208). M.W. gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) through the ESPRIT Programme (Grant No. 156).","quality_controlled":"1","oa_version":"Published Version","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"},{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","grant_number":"716117"},{"name":"Configuration Spaces over Non-Smooth Spaces","_id":"34dbf174-11ca-11ed-8bc3-afe9d43d4b9c","grant_number":"E208"},{"name":"Gradient flow techniques for quantum Markov semigroups","_id":"34c6ea2d-11ca-11ed-8bc3-c04f3c502833","grant_number":"ESP156_N"}],"oa":1,"date_updated":"2023-06-28T11:54:35Z","volume":23,"article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.","lang":"eng"}],"author":[{"last_name":"Dello Schiavo","full_name":"Dello Schiavo, Lorenzo","orcid":"0000-0002-9881-6870","first_name":"Lorenzo","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E"},{"first_name":"Melchior","orcid":"0000-0002-0519-4241","full_name":"Wirth, Melchior","last_name":"Wirth","id":"88644358-0A0E-11EA-8FA5-49A33DDC885E"}],"publication_status":"published","citation":{"ista":"Dello Schiavo L, Wirth M. 2023. Ergodic decompositions of Dirichlet forms under order isomorphisms. Journal of Evolution Equations. 23(1), 9.","short":"L. Dello Schiavo, M. Wirth, Journal of Evolution Equations 23 (2023).","mla":"Dello Schiavo, Lorenzo, and Melchior Wirth. “Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms.” <i>Journal of Evolution Equations</i>, vol. 23, no. 1, 9, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00028-022-00859-7\">10.1007/s00028-022-00859-7</a>.","ama":"Dello Schiavo L, Wirth M. Ergodic decompositions of Dirichlet forms under order isomorphisms. <i>Journal of Evolution Equations</i>. 2023;23(1). doi:<a href=\"https://doi.org/10.1007/s00028-022-00859-7\">10.1007/s00028-022-00859-7</a>","chicago":"Dello Schiavo, Lorenzo, and Melchior Wirth. “Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms.” <i>Journal of Evolution Equations</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00028-022-00859-7\">https://doi.org/10.1007/s00028-022-00859-7</a>.","apa":"Dello Schiavo, L., &#38; Wirth, M. (2023). Ergodic decompositions of Dirichlet forms under order isomorphisms. <i>Journal of Evolution Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00028-022-00859-7\">https://doi.org/10.1007/s00028-022-00859-7</a>","ieee":"L. Dello Schiavo and M. Wirth, “Ergodic decompositions of Dirichlet forms under order isomorphisms,” <i>Journal of Evolution Equations</i>, vol. 23, no. 1. Springer Nature, 2023."},"file":[{"file_id":"12325","creator":"dernst","relation":"main_file","content_type":"application/pdf","success":1,"access_level":"open_access","date_updated":"2023-01-20T10:45:06Z","checksum":"1f34f3e2cb521033de6154f274ea3a4e","date_created":"2023-01-20T10:45:06Z","file_size":422612,"file_name":"2023_JourEvolutionEquations_DelloSchiavo.pdf"}],"date_created":"2023-01-08T23:00:53Z","has_accepted_license":"1","department":[{"_id":"JaMa"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"01","article_type":"original","date_published":"2023-01-01T00:00:00Z","issue":"1","publication":"Journal of Evolution Equations","file_date_updated":"2023-01-20T10:45:06Z","intvolume":"        23","status":"public","day":"01","type":"journal_article"},{"issue":"2","publication":"Journal of Evolution Equations","file_date_updated":"2022-08-16T08:52:46Z","intvolume":"        22","status":"public","day":"01","type":"journal_article","file":[{"creator":"kschuh","file_id":"11862","content_type":"application/pdf","relation":"main_file","success":1,"date_updated":"2022-08-16T08:52:46Z","access_level":"open_access","date_created":"2022-08-16T08:52:46Z","checksum":"59b99d1b48b6bd40983e7ce298524a21","file_name":"2022_Journal of Evolution Equations_Agresti.pdf","file_size":1758371}],"date_created":"2022-08-16T08:39:43Z","has_accepted_license":"1","department":[{"_id":"JuFi"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"06","date_published":"2022-06-01T00:00:00Z","article_type":"original","_id":"11858","publication_identifier":{"eissn":["1424-3202"],"issn":["1424-3199"]},"acknowledgement":"The authors thank Emiel Lorist for helpful comments. The authors thank the anonymous referees for their helpful remarks to improve the presentation.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","quality_controlled":"1","oa_version":"Published Version","oa":1,"volume":22,"date_updated":"2023-08-03T12:53:51Z","article_processing_charge":"Yes (via OA deal)","abstract":[{"lang":"eng","text":"This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical 𝐿2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results."}],"author":[{"first_name":"Antonio","full_name":"Agresti, Antonio","last_name":"Agresti","orcid":"0000-0002-9573-2962","id":"673cd0cc-9b9a-11eb-b144-88f30e1fbb72"},{"last_name":"Veraar","full_name":"Veraar, Mark","first_name":"Mark"}],"keyword":["Mathematics (miscellaneous)"],"publication_status":"published","citation":{"ista":"Agresti A, Veraar M. 2022. Nonlinear parabolic stochastic evolution equations in critical spaces part II. Journal of Evolution Equations. 22(2), 56.","short":"A. Agresti, M. Veraar, Journal of Evolution Equations 22 (2022).","ama":"Agresti A, Veraar M. Nonlinear parabolic stochastic evolution equations in critical spaces part II. <i>Journal of Evolution Equations</i>. 2022;22(2). doi:<a href=\"https://doi.org/10.1007/s00028-022-00786-7\">10.1007/s00028-022-00786-7</a>","mla":"Agresti, Antonio, and Mark Veraar. “Nonlinear Parabolic Stochastic Evolution Equations in Critical Spaces Part II.” <i>Journal of Evolution Equations</i>, vol. 22, no. 2, 56, Springer Nature, 2022, doi:<a href=\"https://doi.org/10.1007/s00028-022-00786-7\">10.1007/s00028-022-00786-7</a>.","chicago":"Agresti, Antonio, and Mark Veraar. “Nonlinear Parabolic Stochastic Evolution Equations in Critical Spaces Part II.” <i>Journal of Evolution Equations</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00028-022-00786-7\">https://doi.org/10.1007/s00028-022-00786-7</a>.","apa":"Agresti, A., &#38; Veraar, M. (2022). Nonlinear parabolic stochastic evolution equations in critical spaces part II. <i>Journal of Evolution Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00028-022-00786-7\">https://doi.org/10.1007/s00028-022-00786-7</a>","ieee":"A. Agresti and M. Veraar, “Nonlinear parabolic stochastic evolution equations in critical spaces part II,” <i>Journal of Evolution Equations</i>, vol. 22, no. 2. Springer Nature, 2022."},"ddc":["510"],"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)"},"article_number":"56","isi":1,"title":"Nonlinear parabolic stochastic evolution equations in critical spaces part II","external_id":{"isi":["000809108500001"]},"year":"2022","doi":"10.1007/s00028-022-00786-7"}]