---
_id: '14934'
abstract:
- lang: eng
  text: "We study random perturbations of a Riemannian manifold (M, g) by means of
    so-called\r\nFractional Gaussian Fields, which are defined intrinsically by the
    given manifold. The fields\r\nh• : ω \x02→ hω will act on the manifold via the
    conformal transformation g \x02→ gω := e2hω g.\r\nOur focus will be on the regular
    case with Hurst parameter H > 0, the critical case H = 0\r\nbeing the celebrated
    Liouville geometry in two dimensions. We want to understand how basic\r\ngeometric
    and functional-analytic quantities like diameter, volume, heat kernel, Brownian\r\nmotion,
    spectral bound, or spectral gap change under the influence of the noise. And if
    so, is\r\nit possible to quantify these dependencies in terms of key parameters
    of the noise? Another\r\ngoal is to define and analyze in detail the Fractional
    Gaussian Fields on a general Riemannian\r\nmanifold, a fascinating object of independent
    interest."
acknowledgement: "The authors would like to thank Matthias Erbar and Ronan Herry for
  valuable discussions on this project. They are also grateful to Nathanaël Berestycki,
  and Fabrice Baudoin for respectively pointing out the references [7], and [6, 24],
  and to Julien Fageot and Thomas Letendre for pointing out a mistake in a previous
  version of the proof of Proposition 3.10. The authors feel very much indebted to
  an anonymous reviewer for his/her careful reading and the many valuable suggestions
  that have significantly contributed to the improvement of the paper. L.D.S. gratefully
  acknowledges financial support by the Deutsche Forschungsgemeinschaft through CRC
  1060 as well as through SPP 2265, and by the Austrian Science Fund (FWF) grant F65
  at Institute of Science and Technology Austria. This research was funded in whole
  or in part by the Austrian Science Fund (FWF) ESPRIT 208. For the purpose of open
  access, the authors have applied a CC BY public copyright licence to any Author
  Accepted Manuscript version arising from this submission. E.K. and K.-T.S. gratefully
  acknowledge funding by the Deutsche Forschungsgemeinschaft through the Hausdorff
  Center for Mathematics and through CRC 1060 as well as through SPP 2265.\r\nOpen
  Access funding enabled and organized by Projekt DEAL."
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: Eva
  full_name: Kopfer, Eva
  last_name: Kopfer
- first_name: Karl Theodor
  full_name: Sturm, Karl Theodor
  last_name: Sturm
citation:
  ama: Dello Schiavo L, Kopfer E, Sturm KT. A discovery tour in random Riemannian
    geometry. <i>Potential Analysis</i>. 2024. doi:<a href="https://doi.org/10.1007/s11118-023-10118-0">10.1007/s11118-023-10118-0</a>
  apa: Dello Schiavo, L., Kopfer, E., &#38; Sturm, K. T. (2024). A discovery tour
    in random Riemannian geometry. <i>Potential Analysis</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s11118-023-10118-0">https://doi.org/10.1007/s11118-023-10118-0</a>
  chicago: Dello Schiavo, Lorenzo, Eva Kopfer, and Karl Theodor Sturm. “A Discovery
    Tour in Random Riemannian Geometry.” <i>Potential Analysis</i>. Springer Nature,
    2024. <a href="https://doi.org/10.1007/s11118-023-10118-0">https://doi.org/10.1007/s11118-023-10118-0</a>.
  ieee: L. Dello Schiavo, E. Kopfer, and K. T. Sturm, “A discovery tour in random
    Riemannian geometry,” <i>Potential Analysis</i>. Springer Nature, 2024.
  ista: Dello Schiavo L, Kopfer E, Sturm KT. 2024. A discovery tour in random Riemannian
    geometry. Potential Analysis.
  mla: Dello Schiavo, Lorenzo, et al. “A Discovery Tour in Random Riemannian Geometry.”
    <i>Potential Analysis</i>, Springer Nature, 2024, doi:<a href="https://doi.org/10.1007/s11118-023-10118-0">10.1007/s11118-023-10118-0</a>.
  short: L. Dello Schiavo, E. Kopfer, K.T. Sturm, Potential Analysis (2024).
date_created: 2024-02-04T23:00:54Z
date_published: 2024-01-26T00:00:00Z
date_updated: 2024-02-05T13:04:23Z
day: '26'
department:
- _id: JaMa
doi: 10.1007/s11118-023-10118-0
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s11118-023-10118-0
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
publication: Potential Analysis
publication_identifier:
  eissn:
  - 1572-929X
  issn:
  - 0926-2601
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: A discovery tour in random Riemannian geometry
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...
---
_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.
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
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'
...