---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '20591'
abstract:
- lang: eng
  text: In this paper we derive estimates for the Hessian of the logarithm (log-Hessian)
    for solutions to the heat equation. For initial data in the form of log-Lipschitz
    perturbation of strongly log-concave measures, the log-Hessian admits an explicit,
    uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant
    of a transport map pushing forward the standard Gaussian to a measure in this
    class. On the other hand, we show that assuming only fast decay of the tails of
    the initial datum does not suffice to guarantee uniform log-Hessian upper bounds.
acknowledgement: This research was funded in part by the Austrian Science Fund (FWF)
  project 10.55776/F65 and by the European Union’s Horizon 2020 research and innovation
  programme under the Marie Sklodowska-Curie grant agreement No 101034413. The authors
  thank Professors Jean Dolbeault, Jan Maas, and Nikita Simonov for many useful comments,
  and Professors Kazuhiro Ishige, Asuka Takatsu, and Yair Shenfeld for inspiring interactions.
article_number: '71'
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Giovanni
  full_name: Brigati, Giovanni
  id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
  last_name: Brigati
- first_name: Francesco
  full_name: Pedrotti, Francesco
  id: d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c
  last_name: Pedrotti
citation:
  ama: Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps.
    <i>Electronic Communications in Probability</i>. 2025;30. doi:<a href="https://doi.org/10.1214/25-ECP717">10.1214/25-ECP717</a>
  apa: Brigati, G., &#38; Pedrotti, F. (2025). Heat flow, log-concavity, and Lipschitz
    transport maps. <i>Electronic Communications in Probability</i>. Institute of
    Mathematical Statistics. <a href="https://doi.org/10.1214/25-ECP717">https://doi.org/10.1214/25-ECP717</a>
  chicago: Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and
    Lipschitz Transport Maps.” <i>Electronic Communications in Probability</i>. Institute
    of Mathematical Statistics, 2025. <a href="https://doi.org/10.1214/25-ECP717">https://doi.org/10.1214/25-ECP717</a>.
  ieee: G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport
    maps,” <i>Electronic Communications in Probability</i>, vol. 30. Institute of
    Mathematical Statistics, 2025.
  ista: Brigati G, Pedrotti F. 2025. Heat flow, log-concavity, and Lipschitz transport
    maps. Electronic Communications in Probability. 30, 71.
  mla: Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz
    Transport Maps.” <i>Electronic Communications in Probability</i>, vol. 30, 71,
    Institute of Mathematical Statistics, 2025, doi:<a href="https://doi.org/10.1214/25-ECP717">10.1214/25-ECP717</a>.
  short: G. Brigati, F. Pedrotti, Electronic Communications in Probability 30 (2025).
corr_author: '1'
date_created: 2025-11-02T23:01:35Z
date_published: 2025-09-25T00:00:00Z
date_updated: 2025-12-01T15:08:54Z
day: '25'
ddc:
- '500'
department:
- _id: JaMa
doi: 10.1214/25-ECP717
ec_funded: 1
external_id:
  arxiv:
  - '2404.15205'
  isi:
  - '001611557000018'
file:
- access_level: open_access
  checksum: 67858edbd74658fe38955fa1216f2f18
  content_type: application/pdf
  creator: dernst
  date_created: 2025-11-04T07:34:05Z
  date_updated: 2025-11-04T07:34:05Z
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  file_name: 2025_ElectronJourProbab_Brigati.pdf
  file_size: 278078
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file_date_updated: 2025-11-04T07:34:05Z
has_accepted_license: '1'
intvolume: '        30'
isi: 1
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Electronic Communications in Probability
publication_identifier:
  eissn:
  - 1083-589X
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
related_material:
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    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Heat flow, log-concavity, and Lipschitz transport maps
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: 30
year: '2025'
...