---
_id: '11354'
abstract:
- lang: eng
  text: We construct a recurrent diffusion process with values in the space of probability
    measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process
    is associated with the Dirichlet form defined by integration of the Wasserstein
    gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional
    base spaces to the modified massive Arratia flow over the unit interval described
    in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800).
    Together with two different constructions of the process, we discuss its ergodicity,
    invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.
acknowledgement: Research supported by the Sonderforschungsbereich 1060 and the Hausdorff
  Center for Mathematics. The author gratefully acknowledges funding of his current
  position at IST Austria by the Austrian Science Fund (FWF) grant F65 and by the
  European Research Council (ERC, Grant agreement No. 716117, awarded to Prof. Dr.
  Jan Maas).
article_processing_charge: No
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. The Dirichlet–Ferguson diffusion on the space of probability
    measures over a closed Riemannian manifold. <i>Annals of Probability</i>. 2022;50(2):591-648.
    doi:<a href="https://doi.org/10.1214/21-AOP1541">10.1214/21-AOP1541</a>
  apa: Dello Schiavo, L. (2022). The Dirichlet–Ferguson diffusion on the space of
    probability measures over a closed Riemannian manifold. <i>Annals of Probability</i>.
    Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/21-AOP1541">https://doi.org/10.1214/21-AOP1541</a>
  chicago: Dello Schiavo, Lorenzo. “The Dirichlet–Ferguson Diffusion on the Space
    of Probability Measures over a Closed Riemannian Manifold.” <i>Annals of Probability</i>.
    Institute of Mathematical Statistics, 2022. <a href="https://doi.org/10.1214/21-AOP1541">https://doi.org/10.1214/21-AOP1541</a>.
  ieee: L. Dello Schiavo, “The Dirichlet–Ferguson diffusion on the space of probability
    measures over a closed Riemannian manifold,” <i>Annals of Probability</i>, vol.
    50, no. 2. Institute of Mathematical Statistics, pp. 591–648, 2022.
  ista: Dello Schiavo L. 2022. The Dirichlet–Ferguson diffusion on the space of probability
    measures over a closed Riemannian manifold. Annals of Probability. 50(2), 591–648.
  mla: Dello Schiavo, Lorenzo. “The Dirichlet–Ferguson Diffusion on the Space of Probability
    Measures over a Closed Riemannian Manifold.” <i>Annals of Probability</i>, vol.
    50, no. 2, Institute of Mathematical Statistics, 2022, pp. 591–648, doi:<a href="https://doi.org/10.1214/21-AOP1541">10.1214/21-AOP1541</a>.
  short: L. Dello Schiavo, Annals of Probability 50 (2022) 591–648.
date_created: 2022-05-08T22:01:44Z
date_published: 2022-03-01T00:00:00Z
date_updated: 2023-10-17T12:50:24Z
day: '01'
department:
- _id: JaMa
doi: 10.1214/21-AOP1541
ec_funded: 1
external_id:
  arxiv:
  - '1811.11598'
  isi:
  - '000773518500005'
intvolume: '        50'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.1811.11598'
month: '03'
oa: 1
oa_version: Preprint
page: 591-648
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
publication: Annals of Probability
publication_identifier:
  eissn:
  - 2168-894X
  issn:
  - 0091-1798
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: The Dirichlet–Ferguson diffusion on the space of probability measures over
  a closed Riemannian manifold
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 50
year: '2022'
...
---
_id: '11418'
abstract:
- lang: eng
  text: "We consider the quadratic form of a general high-rank deterministic matrix
    on the eigenvectors of an N×N\r\nWigner matrix and prove that it has Gaussian
    fluctuation for each bulk eigenvector in the large N limit. The proof is a combination
    of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau
    (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021)
    1005–1048)."
acknowledgement: L.E. would like to thank Zhigang Bao for many illuminating discussions
  in an early stage of this research. The authors are also grateful to Paul Bourgade
  for his comments on the manuscript and the anonymous referee for several useful
  suggestions.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Dominik J
  full_name: Schröder, Dominik J
  id: 408ED176-F248-11E8-B48F-1D18A9856A87
  last_name: Schröder
  orcid: 0000-0002-2904-1856
citation:
  ama: Cipolloni G, Erdös L, Schröder DJ. Normal fluctuation in quantum ergodicity
    for Wigner matrices. <i>Annals of Probability</i>. 2022;50(3):984-1012. doi:<a
    href="https://doi.org/10.1214/21-AOP1552">10.1214/21-AOP1552</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Normal fluctuation
    in quantum ergodicity for Wigner matrices. <i>Annals of Probability</i>. Institute
    of Mathematical Statistics. <a href="https://doi.org/10.1214/21-AOP1552">https://doi.org/10.1214/21-AOP1552</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Normal Fluctuation
    in Quantum Ergodicity for Wigner Matrices.” <i>Annals of Probability</i>. Institute
    of Mathematical Statistics, 2022. <a href="https://doi.org/10.1214/21-AOP1552">https://doi.org/10.1214/21-AOP1552</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Normal fluctuation in quantum
    ergodicity for Wigner matrices,” <i>Annals of Probability</i>, vol. 50, no. 3.
    Institute of Mathematical Statistics, pp. 984–1012, 2022.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2022. Normal fluctuation in quantum ergodicity
    for Wigner matrices. Annals of Probability. 50(3), 984–1012.
  mla: Cipolloni, Giorgio, et al. “Normal Fluctuation in Quantum Ergodicity for Wigner
    Matrices.” <i>Annals of Probability</i>, vol. 50, no. 3, Institute of Mathematical
    Statistics, 2022, pp. 984–1012, doi:<a href="https://doi.org/10.1214/21-AOP1552">10.1214/21-AOP1552</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, Annals of Probability 50 (2022) 984–1012.
date_created: 2022-05-29T22:01:53Z
date_published: 2022-05-01T00:00:00Z
date_updated: 2023-08-03T07:16:53Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/21-AOP1552
external_id:
  arxiv:
  - '2103.06730'
  isi:
  - '000793963400005'
intvolume: '        50'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2103.06730
month: '05'
oa: 1
oa_version: Preprint
page: 984-1012
publication: Annals of Probability
publication_identifier:
  eissn:
  - 2168-894X
  issn:
  - 0091-1798
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Normal fluctuation in quantum ergodicity for Wigner matrices
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 50
year: '2022'
...