---
_id: '12104'
abstract:
- lang: eng
  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.
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).
article_number: '9'
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
- first_name: Melchior
  full_name: Wirth, Melchior
  id: 88644358-0A0E-11EA-8FA5-49A33DDC885E
  last_name: Wirth
  orcid: 0000-0002-0519-4241
citation:
  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>
  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>
  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>.
  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.
  ista: Dello Schiavo L, Wirth M. 2023. Ergodic decompositions of Dirichlet forms
    under order isomorphisms. Journal of Evolution Equations. 23(1), 9.
  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>.
  short: L. Dello Schiavo, M. Wirth, Journal of Evolution Equations 23 (2023).
date_created: 2023-01-08T23:00:53Z
date_published: 2023-01-01T00:00:00Z
date_updated: 2023-06-28T11:54:35Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00028-022-00859-7
ec_funded: 1
external_id:
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  - '000906214600004'
file:
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language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
project:
- _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
- _id: 34dbf174-11ca-11ed-8bc3-afe9d43d4b9c
  grant_number: E208
  name: Configuration Spaces over Non-Smooth Spaces
- _id: 34c6ea2d-11ca-11ed-8bc3-c04f3c502833
  grant_number: ESP156_N
  name: Gradient flow techniques for quantum Markov semigroups
publication: Journal of Evolution Equations
publication_identifier:
  eissn:
  - 1424-3202
  issn:
  - 1424-3199
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Ergodic decompositions of Dirichlet forms under order isomorphisms
tmp:
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 23
year: '2023'
...
---
_id: '11858'
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 \U0001D43F2-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."
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)."
article_number: '56'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Antonio
  full_name: Agresti, Antonio
  id: 673cd0cc-9b9a-11eb-b144-88f30e1fbb72
  last_name: Agresti
  orcid: 0000-0002-9573-2962
- first_name: Mark
  full_name: Veraar, Mark
  last_name: Veraar
citation:
  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>
  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>
  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>.
  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.
  ista: Agresti A, Veraar M. 2022. Nonlinear parabolic stochastic evolution equations
    in critical spaces part II. Journal of Evolution Equations. 22(2), 56.
  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>.
  short: A. Agresti, M. Veraar, Journal of Evolution Equations 22 (2022).
date_created: 2022-08-16T08:39:43Z
date_published: 2022-06-01T00:00:00Z
date_updated: 2023-08-03T12:53:51Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00028-022-00786-7
external_id:
  isi:
  - '000809108500001'
file:
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  date_created: 2022-08-16T08:52:46Z
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  file_name: 2022_Journal of Evolution Equations_Agresti.pdf
  file_size: 1758371
  relation: main_file
  success: 1
file_date_updated: 2022-08-16T08:52:46Z
has_accepted_license: '1'
intvolume: '        22'
isi: 1
issue: '2'
keyword:
- Mathematics (miscellaneous)
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: Journal of Evolution Equations
publication_identifier:
  eissn:
  - 1424-3202
  issn:
  - 1424-3199
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Nonlinear parabolic stochastic evolution equations in critical spaces part
  II
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 22
year: '2022'
...