---
_id: '12485'
abstract:
- lang: eng
  text: In this paper we introduce the critical variational setting for parabolic
    stochastic evolution equations of quasi- or semi-linear type. Our results improve
    many of the abstract results in the classical variational setting. In particular,
    we are able to replace the usual weak or local monotonicity condition by a more
    flexible local Lipschitz condition. Moreover, the usual growth conditions on the
    multiplicative noise are weakened considerably. Our new setting provides general
    conditions under which local and global existence and uniqueness hold. Moreover,
    we prove continuous dependence on the initial data. We show that many classical
    SPDEs, which could not be covered by the classical variational setting, do fit
    in the critical variational setting. In particular, this is the case for the Cahn-Hilliard
    equations, tamed Navier-Stokes equations, and Allen-Cahn equation.
acknowledgement: The first author has received funding from the European Research
  Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (grant agreement No 948819) . The second author is supported by the VICI subsidy
  VI.C.212.027 of the Netherlands Organisation for Scientific Research (NWO).
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
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. The critical variational setting for stochastic evolution
    equations. <i>Probability Theory and Related Fields</i>. 2024;188:957-1015. doi:<a
    href="https://doi.org/10.1007/s00440-023-01249-x">10.1007/s00440-023-01249-x</a>
  apa: Agresti, A., &#38; Veraar, M. (2024). The critical variational setting for
    stochastic evolution equations. <i>Probability Theory and Related Fields</i>.
    Springer Nature. <a href="https://doi.org/10.1007/s00440-023-01249-x">https://doi.org/10.1007/s00440-023-01249-x</a>
  chicago: Agresti, Antonio, and Mark Veraar. “The Critical Variational Setting for
    Stochastic Evolution Equations.” <i>Probability Theory and Related Fields</i>.
    Springer Nature, 2024. <a href="https://doi.org/10.1007/s00440-023-01249-x">https://doi.org/10.1007/s00440-023-01249-x</a>.
  ieee: A. Agresti and M. Veraar, “The critical variational setting for stochastic
    evolution equations,” <i>Probability Theory and Related Fields</i>, vol. 188.
    Springer Nature, pp. 957–1015, 2024.
  ista: Agresti A, Veraar M. 2024. The critical variational setting for stochastic
    evolution equations. Probability Theory and Related Fields. 188, 957–1015.
  mla: Agresti, Antonio, and Mark Veraar. “The Critical Variational Setting for Stochastic
    Evolution Equations.” <i>Probability Theory and Related Fields</i>, vol. 188,
    Springer Nature, 2024, pp. 957–1015, doi:<a href="https://doi.org/10.1007/s00440-023-01249-x">10.1007/s00440-023-01249-x</a>.
  short: A. Agresti, M. Veraar, Probability Theory and Related Fields 188 (2024) 957–1015.
date_created: 2023-02-02T10:45:15Z
date_published: 2024-04-01T00:00:00Z
date_updated: 2025-09-04T11:27:46Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00440-023-01249-x
ec_funded: 1
external_id:
  arxiv:
  - '2206.00230'
  isi:
  - '001154226500001'
file:
- access_level: open_access
  checksum: b8572339dbc5b8de4934dc5fd34afc7d
  content_type: application/pdf
  creator: dernst
  date_created: 2024-07-22T09:21:09Z
  date_updated: 2024-07-22T09:21:09Z
  file_id: '17296'
  file_name: 2024_ProbTheory_Agresti.pdf
  file_size: 942801
  relation: main_file
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file_date_updated: 2024-07-22T09:21:09Z
fulldoi: https://doi.org/10.1007/s00440-023-01249-x
has_accepted_license: '1'
intvolume: '       188'
isi: 1
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 957-1015
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
  call_identifier: H2020
  grant_number: '948819'
  name: Bridging Scales in Random Materials
publication: Probability Theory and Related Fields
publication_identifier:
  eissn:
  - 1432-2064
  issn:
  - 0178-8051
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The critical variational setting for stochastic evolution equations
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 188
year: '2024'
...