--- _id: '10145' abstract: - lang: eng text: We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique. acknowledgement: The author is grateful to Professors Sergio Albeverio and Andreas Eberle, and to Dr. Kohei Suzuki, for fruitful conversations on the subject of the present work, and for respectively pointing out the references [1, 13], and [3, 20]. Finally, he is especially grateful to an anonymous Reviewer for their very careful reading and their suggestions which improved the readability of the paper. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Lorenzo full_name: Dello Schiavo, Lorenzo id: ECEBF480-9E4F-11EA-B557-B0823DDC885E last_name: Dello Schiavo orcid: 0000-0002-9881-6870 citation: ama: Dello Schiavo L. Ergodic decomposition of Dirichlet forms via direct integrals and applications. Potential Analysis. 2023;58:573-615. doi:10.1007/s11118-021-09951-y apa: Dello Schiavo, L. (2023). Ergodic decomposition of Dirichlet forms via direct integrals and applications. Potential Analysis. Springer Nature. https://doi.org/10.1007/s11118-021-09951-y chicago: Dello Schiavo, Lorenzo. “Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications.” Potential Analysis. Springer Nature, 2023. https://doi.org/10.1007/s11118-021-09951-y. ieee: L. Dello Schiavo, “Ergodic decomposition of Dirichlet forms via direct integrals and applications,” Potential Analysis, vol. 58. Springer Nature, pp. 573–615, 2023. ista: Dello Schiavo L. 2023. Ergodic decomposition of Dirichlet forms via direct integrals and applications. Potential Analysis. 58, 573–615. mla: Dello Schiavo, Lorenzo. “Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications.” Potential Analysis, vol. 58, Springer Nature, 2023, pp. 573–615, doi:10.1007/s11118-021-09951-y. short: L. Dello Schiavo, Potential Analysis 58 (2023) 573–615. date_created: 2021-10-17T22:01:17Z date_published: 2023-03-01T00:00:00Z date_updated: 2023-10-04T09:19:12Z day: '01' ddc: - '510' department: - _id: JaMa doi: 10.1007/s11118-021-09951-y ec_funded: 1 external_id: arxiv: - '2003.01366' isi: - '000704213400001' file: - access_level: open_access checksum: 625526482be300ca7281c91c30d41725 content_type: application/pdf creator: dernst date_created: 2023-10-04T09:18:59Z date_updated: 2023-10-04T09:18:59Z file_id: '14387' file_name: 2023_PotentialAnalysis_DelloSchiavo.pdf file_size: 806391 relation: main_file success: 1 file_date_updated: 2023-10-04T09:18:59Z has_accepted_license: '1' intvolume: ' 58' isi: 1 language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '03' oa: 1 oa_version: Published Version page: 573-615 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics publication: Potential Analysis publication_identifier: eissn: - 1572-929X issn: - 0926-2601 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Ergodic decomposition of Dirichlet forms via direct integrals and applications 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: 58 year: '2023' ...