---
_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
arxiv: 1
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. <i>Potential Analysis</i>. 2023;58:573-615. doi:<a href="https://doi.org/10.1007/s11118-021-09951-y">10.1007/s11118-021-09951-y</a>
  apa: Dello Schiavo, L. (2023). Ergodic decomposition of Dirichlet forms via direct
    integrals and applications. <i>Potential Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s11118-021-09951-y">https://doi.org/10.1007/s11118-021-09951-y</a>
  chicago: Dello Schiavo, Lorenzo. “Ergodic Decomposition of Dirichlet Forms via Direct
    Integrals and Applications.” <i>Potential Analysis</i>. Springer Nature, 2023.
    <a href="https://doi.org/10.1007/s11118-021-09951-y">https://doi.org/10.1007/s11118-021-09951-y</a>.
  ieee: L. Dello Schiavo, “Ergodic decomposition of Dirichlet forms via direct integrals
    and applications,” <i>Potential Analysis</i>, 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.” <i>Potential Analysis</i>, vol. 58, Springer Nature,
    2023, pp. 573–615, doi:<a href="https://doi.org/10.1007/s11118-021-09951-y">10.1007/s11118-021-09951-y</a>.
  short: L. Dello Schiavo, Potential Analysis 58 (2023) 573–615.
corr_author: '1'
date_created: 2021-10-17T22:01:17Z
date_published: 2023-03-01T00:00:00Z
date_updated: 2025-04-14T07:27:46Z
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
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  file_name: 2023_PotentialAnalysis_DelloSchiavo.pdf
  file_size: 806391
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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'
...