---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21018'
abstract:
- lang: eng
text: In this paper, we review recent results on stability and instability in logarithmic
Sobolev inequalities, with a particular emphasis on strong norms. We consider
several versions of these inequalities on the Euclidean space, for the Lebesgue
and the Gaussian measures, and discuss their differences in terms of moments and
stability. We give new and direct proofs, as well as examples and discuss the
stability of a logarithmic uncertainty principle. Although we do not cover all
aspects of the topic, we hope to contribute to establishing the state of the art.
acknowledgement: This work has been supported by the Project Conviviality (ANR-23-CE40–0003)
of the French National Research Agency. G.B. has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No 101034413. The authors thank a referee for a careful reading
and suggestions which result in a significant improvement of the manuscript. Open
access funding provided by Institute of Science and Technology (IST Austria). The
work of GB has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413.
This work has been supported by the Project Conviviality (ANR-23-CE40–0003) of the
French National Research Agency.
article_number: '5'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Jean
full_name: Dolbeault, Jean
last_name: Dolbeault
- first_name: Nikita
full_name: Simonov, Nikita
last_name: Simonov
citation:
ama: 'Brigati G, Dolbeault J, Simonov N. Logarithmic Sobolev Inequalities: A review
on stability and instability results. La Matematica. 2026;5. doi:10.1007/s44007-025-00180-y'
apa: 'Brigati, G., Dolbeault, J., & Simonov, N. (2026). Logarithmic Sobolev
Inequalities: A review on stability and instability results. La Matematica.
Springer Nature. https://doi.org/10.1007/s44007-025-00180-y'
chicago: 'Brigati, Giovanni, Jean Dolbeault, and Nikita Simonov. “Logarithmic Sobolev
Inequalities: A Review on Stability and Instability Results.” La Matematica.
Springer Nature, 2026. https://doi.org/10.1007/s44007-025-00180-y.'
ieee: 'G. Brigati, J. Dolbeault, and N. Simonov, “Logarithmic Sobolev Inequalities:
A review on stability and instability results,” La Matematica, vol. 5.
Springer Nature, 2026.'
ista: 'Brigati G, Dolbeault J, Simonov N. 2026. Logarithmic Sobolev Inequalities:
A review on stability and instability results. La Matematica. 5, 5.'
mla: 'Brigati, Giovanni, et al. “Logarithmic Sobolev Inequalities: A Review on Stability
and Instability Results.” La Matematica, vol. 5, 5, Springer Nature, 2026,
doi:10.1007/s44007-025-00180-y.'
short: G. Brigati, J. Dolbeault, N. Simonov, La Matematica 5 (2026).
corr_author: '1'
date_created: 2026-01-20T10:14:55Z
date_published: 2026-01-08T00:00:00Z
date_updated: 2026-01-21T07:48:28Z
day: '08'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s44007-025-00180-y
ec_funded: 1
external_id:
arxiv:
- '2504.08658'
file:
- access_level: open_access
checksum: 0702d8397f216555b1d5286e5d77f09c
content_type: application/pdf
creator: dernst
date_created: 2026-01-21T07:45:03Z
date_updated: 2026-01-21T07:45:03Z
file_id: '21025'
file_name: 2026_LaMatematica_Brigati.pdf
file_size: 4992025
relation: main_file
success: 1
file_date_updated: 2026-01-21T07:45:03Z
has_accepted_license: '1'
intvolume: ' 5'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: La Matematica
publication_identifier:
issn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Logarithmic Sobolev Inequalities: A review on stability and instability results'
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: 5
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21881'
abstract:
- lang: eng
text: "I review recent contributions on nonlinear Dirichlet forms. Then, I specialise
to the case of 2-\r\nhomogeneous and local forms. Inspired by the theory of Finsler
manifolds and metric measure spaces, I establish new properties of such nonlinear
Dirichlet forms, which are reminiscent of differential calculus formulae."
acknowledgement: I am thankful to G. Savaré, for introducing me to the study of nonlinear
Dirichlet forms and metric measure spaces, and to D. Manini for stimulating discussions.
Open access funding provided by Institute of Science and Technology (IST Austria).
The author has been funded by the European Union’s Horizon 2020 research and innovation
program under the Marie Skłodowska-Curie grant agreement No 754362. Partial support
has been obtained from the EFI ANR-17-CE40-0030 Project of the French National Research
Agency.
article_number: '33'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
citation:
ama: Brigati G. Nonlinear Dirichlet forms, energy spaces, and calculus rules. La
Matematica. 2026;5(2). doi:10.1007/s44007-026-00217-w
apa: Brigati, G. (2026). Nonlinear Dirichlet forms, energy spaces, and calculus
rules. La Matematica. Springer Nature. https://doi.org/10.1007/s44007-026-00217-w
chicago: Brigati, Giovanni. “Nonlinear Dirichlet Forms, Energy Spaces, and Calculus
Rules.” La Matematica. Springer Nature, 2026. https://doi.org/10.1007/s44007-026-00217-w.
ieee: G. Brigati, “Nonlinear Dirichlet forms, energy spaces, and calculus rules,”
La Matematica, vol. 5, no. 2. Springer Nature, 2026.
ista: Brigati G. 2026. Nonlinear Dirichlet forms, energy spaces, and calculus rules.
La Matematica. 5(2), 33.
mla: Brigati, Giovanni. “Nonlinear Dirichlet Forms, Energy Spaces, and Calculus
Rules.” La Matematica, vol. 5, no. 2, 33, Springer Nature, 2026, doi:10.1007/s44007-026-00217-w.
short: G. Brigati, La Matematica 5 (2026).
corr_author: '1'
date_created: 2026-05-17T22:02:10Z
date_published: 2026-04-29T00:00:00Z
date_updated: 2026-05-18T08:27:08Z
day: '29'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s44007-026-00217-w
external_id:
arxiv:
- '2309.00377'
file:
- access_level: open_access
checksum: f75bffe3c793a2cbb26b8494024d0681
content_type: application/pdf
creator: dernst
date_created: 2026-05-18T08:26:15Z
date_updated: 2026-05-18T08:26:15Z
file_id: '21892'
file_name: 2026_LaMathematica_Brigati.pdf
file_size: 394082
relation: main_file
success: 1
file_date_updated: 2026-05-18T08:26:15Z
has_accepted_license: '1'
intvolume: ' 5'
issue: '2'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
publication: La Matematica
publication_identifier:
eissn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Nonlinear Dirichlet forms, energy spaces, and calculus rules
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: 5
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21504'
abstract:
- lang: eng
text: Selecting an appropriate divergence measure is a critical aspect of machine
learning, as it directly impacts model performance. Among the most widely used,
we find the Kullback–Leibler (KL) divergence, originally introduced in kinetic
theory as a measure of relative entropy between probability distributions. Just
as in machine learning, the ability to quantify the proximity of probability distributions
plays a central role in kinetic theory. In this paper, we present a comparative
review of divergence measures rooted in kinetic theory, highlighting their theoretical
foundations and exploring their potential applications in machine learning and
artificial intelligence.
acknowledgement: "This work has been written within the activities of GNCS and GNFM
groups of INdAM (Italian\r\nNational Institute of High Mathematics). G.B. has been
funded by the European Union’s Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreement No 101034413. P.G. has been funded by
the European Union - NextGenerationEU, in the framework of the GRINSGrowing Resilient,
INclusive and Sustainable (GRINS PE00000018)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gennaro
full_name: Auricchio, Gennaro
last_name: Auricchio
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Paolo
full_name: Giudici, Paolo
last_name: Giudici
- first_name: Giuseppe
full_name: Toscani, Giuseppe
last_name: Toscani
citation:
ama: 'Auricchio G, Brigati G, Giudici P, Toscani G. From kinetic theory to AI: A
rediscovery of high-dimensional divergences and their properties. Mathematical
Models and Methods in Applied Sciences. 2026;36(6):1185-1233. doi:10.1142/S0218202526410010'
apa: 'Auricchio, G., Brigati, G., Giudici, P., & Toscani, G. (2026). From kinetic
theory to AI: A rediscovery of high-dimensional divergences and their properties.
Mathematical Models and Methods in Applied Sciences. World Scientific Publishing.
https://doi.org/10.1142/S0218202526410010'
chicago: 'Auricchio, Gennaro, Giovanni Brigati, Paolo Giudici, and Giuseppe Toscani.
“From Kinetic Theory to AI: A Rediscovery of High-Dimensional Divergences and
Their Properties.” Mathematical Models and Methods in Applied Sciences.
World Scientific Publishing, 2026. https://doi.org/10.1142/S0218202526410010.'
ieee: 'G. Auricchio, G. Brigati, P. Giudici, and G. Toscani, “From kinetic theory
to AI: A rediscovery of high-dimensional divergences and their properties,” Mathematical
Models and Methods in Applied Sciences, vol. 36, no. 6. World Scientific Publishing,
pp. 1185–1233, 2026.'
ista: 'Auricchio G, Brigati G, Giudici P, Toscani G. 2026. From kinetic theory to
AI: A rediscovery of high-dimensional divergences and their properties. Mathematical
Models and Methods in Applied Sciences. 36(6), 1185–1233.'
mla: 'Auricchio, Gennaro, et al. “From Kinetic Theory to AI: A Rediscovery of High-Dimensional
Divergences and Their Properties.” Mathematical Models and Methods in Applied
Sciences, vol. 36, no. 6, World Scientific Publishing, 2026, pp. 1185–233,
doi:10.1142/S0218202526410010.'
short: G. Auricchio, G. Brigati, P. Giudici, G. Toscani, Mathematical Models and
Methods in Applied Sciences 36 (2026) 1185–1233.
das_tickbox: '0'
date_created: 2026-03-29T22:07:08Z
date_published: 2026-06-01T00:00:00Z
date_updated: 2026-07-27T12:11:38Z
day: '01'
department:
- _id: JaMa
doi: 10.1142/S0218202526410010
ec_funded: 1
external_id:
arxiv:
- '2507.11387'
intvolume: ' 36'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2507.11387
mathsc:
- 35B40
- 35L60
- 35K55
- 35Q70
- 35Q91
- 35Q92
month: '06'
oa: 1
oa_version: Preprint
page: 1185-1233
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
eissn:
- 1793-6314
issn:
- 0218-2025
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: 'From kinetic theory to AI: A rediscovery of high-dimensional divergences and
their properties'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 36
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21132'
abstract:
- lang: eng
text: We unify the variational hypocoercivity framework established by D. Albritton,
S. Armstrong, J.-C. Mourrat, and M. Novack [2], with the notion of second-order
lifts of reversible diffusion processes, recently introduced by A. Eberle and
the second author [30]. We give an abstract, yet fully constructive, presentation
of the theory, so that it can be applied to a large class of linear kinetic equations.
As this hypocoercivity technique does not twist the reference norm, we can recover
accurate and sharp convergence rates in various models. Among those, adaptive
Langevin dynamics (ALD) is discussed in full detail and we show that for near-quadratic
potentials, with suitable choices of parameters, it is a near-optimal second-order
lift of the overdamped Langevin dynamics. As a further consequence, we observe
that the Generalised Langevin Equation (GLE) is also a second-order lift, as the
standard (kinetic) Langevin dynamics are, of the overdamped Langevin dynamics.
Then, convergence of (GLE) cannot exceed ballistic speed, i.e. the square root
of the rate of the overdamped regime. We illustrate this phenomenon with explicit
computations in a benchmark Gaussian case.
acknowledgement: "We would like to thank Andreas Eberle and Gabriel Stoltz for many
helpful discussions. GB\r\nhas received funding from the European Union Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement
No 101034413. FL wurde gefördert durch die Deutsche Forschungsgemeinschaft (DFG)
im Rahmen der Exzellenzstrategie des Bundes und der Länder – GZ2047/1, Projekt-ID
390685813. LW is supported by the National Science Foundation via grant DMS-2407166.
He is also indebted to the Mathematical Sciences department at Carnegie Mellon University
for partly supporting his visit to Europe in July 2024. Part of this work was completed
when GB and LW were visiting the Institute for Applied Mathematics in Bonn. GB and
LW would like to thank IAM for their hospitality."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Francis
full_name: Lörler, Francis
last_name: Lörler
- first_name: Lihan
full_name: Wang, Lihan
last_name: Wang
citation:
ama: Brigati G, Lörler F, Wang L. Hypocoercivity meets lifts. Kinetic and Related
Models. 2026;20:34-55. doi:10.3934/krm.2025020
apa: Brigati, G., Lörler, F., & Wang, L. (2026). Hypocoercivity meets lifts.
Kinetic and Related Models. AIMS. https://doi.org/10.3934/krm.2025020
chicago: Brigati, Giovanni, Francis Lörler, and Lihan Wang. “Hypocoercivity Meets
Lifts.” Kinetic and Related Models. AIMS, 2026. https://doi.org/10.3934/krm.2025020.
ieee: G. Brigati, F. Lörler, and L. Wang, “Hypocoercivity meets lifts,” Kinetic
and Related Models, vol. 20. AIMS, pp. 34–55, 2026.
ista: Brigati G, Lörler F, Wang L. 2026. Hypocoercivity meets lifts. Kinetic and
Related Models. 20, 34–55.
mla: Brigati, Giovanni, et al. “Hypocoercivity Meets Lifts.” Kinetic and Related
Models, vol. 20, AIMS, 2026, pp. 34–55, doi:10.3934/krm.2025020.
short: G. Brigati, F. Lörler, L. Wang, Kinetic and Related Models 20 (2026) 34–55.
das_tickbox: '1'
date_created: 2026-02-01T23:01:43Z
date_published: 2026-04-01T00:00:00Z
date_updated: 2026-08-12T06:20:09Z
day: '01'
department:
- _id: JaMa
doi: 10.3934/krm.2025020
ec_funded: 1
external_id:
arxiv:
- '2412.10890'
intvolume: ' 20'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2412.10890
month: '04'
oa: 1
oa_version: Preprint
page: 34-55
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Kinetic and Related Models
publication_identifier:
eissn:
- 1937-5077
issn:
- 1937-5093
publication_status: published
publisher: AIMS
quality_controlled: '1'
scopus_import: '1'
status: public
title: Hypocoercivity meets lifts
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '19565'
abstract:
- lang: eng
text: 'Measuring distances in a multidimensional setting is a challenging problem,
which appears in many fields of science and engineering. In this paper, to measure
the distance between two multivariate distributions, we introduce a new measure
of discrepancy which is scale invariant and which, in the case of two independent
copies of the same distribution, and after normalization, coincides with the scaling
invariant multidimensional version of the Gini index recently proposed in [P.
Giudici, E. Raffinetti and G. Toscani, Measuring multidimensional inequality:
A new proposal based on the Fourier transform, preprint (2024), arXiv:2401.14012
]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained
by application of the theory to a pair of Gaussian multidimensional densities.
The obtained metric does improve the standard metrics, based on the mean squared
error, as it is scale invariant. The importance of this theoretical finding is
illustrated by means of a real problem that concerns measuring the importance
of Environmental, Social and Governance factors for the growth of small and medium
enterprises. '
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gennaro
full_name: Auricchio, Gennaro
last_name: Auricchio
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Paolo
full_name: Giudici, Paolo
last_name: Giudici
- first_name: Giuseppe
full_name: Toscani, Giuseppe
last_name: Toscani
citation:
ama: Auricchio G, Brigati G, Giudici P, Toscani G. Multivariate Gini-type discrepancies.
Mathematical Models and Methods in Applied Sciences. 2025;35(5):1267-1296.
doi:10.1142/s0218202525500174
apa: Auricchio, G., Brigati, G., Giudici, P., & Toscani, G. (2025). Multivariate
Gini-type discrepancies. Mathematical Models and Methods in Applied Sciences.
World Scientific Publishing. https://doi.org/10.1142/s0218202525500174
chicago: Auricchio, Gennaro, Giovanni Brigati, Paolo Giudici, and Giuseppe Toscani.
“Multivariate Gini-Type Discrepancies.” Mathematical Models and Methods in
Applied Sciences. World Scientific Publishing, 2025. https://doi.org/10.1142/s0218202525500174.
ieee: G. Auricchio, G. Brigati, P. Giudici, and G. Toscani, “Multivariate Gini-type
discrepancies,” Mathematical Models and Methods in Applied Sciences, vol.
35, no. 5. World Scientific Publishing, pp. 1267–1296, 2025.
ista: Auricchio G, Brigati G, Giudici P, Toscani G. 2025. Multivariate Gini-type
discrepancies. Mathematical Models and Methods in Applied Sciences. 35(5), 1267–1296.
mla: Auricchio, Gennaro, et al. “Multivariate Gini-Type Discrepancies.” Mathematical
Models and Methods in Applied Sciences, vol. 35, no. 5, World Scientific Publishing,
2025, pp. 1267–96, doi:10.1142/s0218202525500174.
short: G. Auricchio, G. Brigati, P. Giudici, G. Toscani, Mathematical Models and
Methods in Applied Sciences 35 (2025) 1267–1296.
date_created: 2025-04-15T13:34:00Z
date_published: 2025-05-01T00:00:00Z
date_updated: 2025-09-30T11:36:56Z
day: '01'
department:
- _id: JaMa
doi: 10.1142/s0218202525500174
external_id:
arxiv:
- '2411.01052'
isi:
- '001456337300001'
intvolume: ' 35'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2411.01052
month: '05'
oa: 1
oa_version: Preprint
page: 1267-1296
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
eissn:
- 1793-6314
issn:
- 0218-2025
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Multivariate Gini-type discrepancies
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 35
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20155'
abstract:
- lang: eng
text: We study time averages for the norm of solutions to kinetic Fokker–Planck
equations associated with general Hamiltonians. We provide fully explicit and
constructive decay estimates for systems subject to a confining potential, allowing
fat-tail, subexponential and (super-)exponential local equilibria, which also
include the classic Maxwellian case. The key step in our estimates is a modified
Poincaré inequality, obtained via a Lions–Poincaré inequality and an averaging
lemma.
acknowledgement: The first author was funded by the European Union's Horizon 2020
research andinnovation program under the Marie Sklodowska-Curie grant agreements
754362 and 101034413,and partially by Project EFI (ANR-17-CE40-0030) of the French
National Research Agency (ANR).The work of the second author was partially funded
by the European Research Council (ERC) underthe European Union's Horizon 2020 research
and innovation programme (grant agreement 810367),and by the Agence Nationale de
la Recherche under grants ANR-19-CE40-0010 (QuAMProcs) andANR-21-CE40-0006 (SINEQ).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Gabriel
full_name: Stoltz, Gabriel
last_name: Stoltz
citation:
ama: Brigati G, Stoltz G. How to construct explicit decay rates for kinetic Fokker–Planck
equations? SIAM Journal on Mathematical Analysis. 2025;57(4):3587-3622.
doi:10.1137/24M1700351
apa: Brigati, G., & Stoltz, G. (2025). How to construct explicit decay rates
for kinetic Fokker–Planck equations? SIAM Journal on Mathematical Analysis.
Society for Industrial and Applied Mathematics. https://doi.org/10.1137/24M1700351
chicago: Brigati, Giovanni, and Gabriel Stoltz. “How to Construct Explicit Decay
Rates for Kinetic Fokker–Planck Equations?” SIAM Journal on Mathematical Analysis.
Society for Industrial and Applied Mathematics, 2025. https://doi.org/10.1137/24M1700351.
ieee: G. Brigati and G. Stoltz, “How to construct explicit decay rates for kinetic
Fokker–Planck equations?,” SIAM Journal on Mathematical Analysis, vol.
57, no. 4. Society for Industrial and Applied Mathematics, pp. 3587–3622, 2025.
ista: Brigati G, Stoltz G. 2025. How to construct explicit decay rates for kinetic
Fokker–Planck equations? SIAM Journal on Mathematical Analysis. 57(4), 3587–3622.
mla: Brigati, Giovanni, and Gabriel Stoltz. “How to Construct Explicit Decay Rates
for Kinetic Fokker–Planck Equations?” SIAM Journal on Mathematical Analysis,
vol. 57, no. 4, Society for Industrial and Applied Mathematics, 2025, pp. 3587–622,
doi:10.1137/24M1700351.
short: G. Brigati, G. Stoltz, SIAM Journal on Mathematical Analysis 57 (2025) 3587–3622.
corr_author: '1'
date_created: 2025-08-10T22:01:29Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-11-05T13:51:40Z
day: '01'
department:
- _id: JaMa
doi: 10.1137/24M1700351
ec_funded: 1
external_id:
arxiv:
- '2302.14506'
isi:
- '001550830900006'
intvolume: ' 57'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2302.14506
month: '08'
oa: 1
oa_version: Preprint
page: 3587-3622
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
eissn:
- 1095-7154
issn:
- 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: How to construct explicit decay rates for kinetic Fokker–Planck equations?
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 57
year: '2025'
...
---
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.
Electronic Communications in Probability. 2025;30. doi:10.1214/25-ECP717
apa: Brigati, G., & Pedrotti, F. (2025). Heat flow, log-concavity, and Lipschitz
transport maps. Electronic Communications in Probability. Institute of
Mathematical Statistics. https://doi.org/10.1214/25-ECP717
chicago: Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and
Lipschitz Transport Maps.” Electronic Communications in Probability. Institute
of Mathematical Statistics, 2025. https://doi.org/10.1214/25-ECP717.
ieee: G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport
maps,” Electronic Communications in Probability, 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.” Electronic Communications in Probability, vol. 30, 71,
Institute of Mathematical Statistics, 2025, doi:10.1214/25-ECP717.
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
file_id: '20596'
file_name: 2025_ElectronJourProbab_Brigati.pdf
file_size: 278078
relation: main_file
success: 1
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:
record:
- id: '17353'
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'
...
---
OA_place: repository
OA_type: green
_id: '20569'
abstract:
- lang: eng
text: 'This is the first part of a general description in terms of mass transport
for time-evolving interacting particles systems, at a mesoscopic level. Beyond
kinetic theory, our framework naturally applies in biology, computer vision, and
engineering. The central object of our study is a new discrepancy d between two
probability distributions in position and velocity states, which is reminiscent
of the 2-Wasserstein distance, but of second-order nature. We construct d in two
steps. First, we optimise over transport plans. The cost function is given by
the minimal acceleration between two coupled states on a fixed time horizon T.
Second, we further optimise over the time horizon T > 0. We prove the existence
of optimal transport plans and maps, and study two time-continuous characterisations
of d. One is given in terms of dynamical transport plans. The other one -- in
the spirit of the Benamou--Brenier formula -- is formulated as the minimisation
of an action of the acceleration field, constrained by Vlasov''s equations. Equivalence
of static and dynamical formulations of d holds true. While part of this result
can be derived from recent, parallel developments in optimal control between measures,
we give an original proof relying on two new ingredients: Galilean regularisation
of Vlasov''s equations and a kinetic Monge--Mather shortening principle. Finally,
we establish a first-order differential calculus in the geometry induced by d,
and identify solutions to Vlasov''s equations with curves of measures satisfying
a certain d-absolute continuity condition. One consequence is an explicit formula
for the d-derivative of such curves.'
acknowledgement: "This work was partially inspired by an unpublished note from 2014
by Guillaume Carlier,\r\nJean Dolbeault, and Bruno Nazaret. GB deeply thanks Jean
Dolbeault for proposing\r\nthis problem to him, guiding him into the subject, and
sharing the aforementioned note.\r\nWe are grateful to Karthik Elamvazhuthi for
making us aware of the work [20].\r\nThe work of GB has received funding from the
European Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement\r\nNo 101034413.\r\nJM and FQ gratefully acknowledge
support from the Austrian Science Fund (FWF)\r\nproject 10.55776/F65."
article_number: '2502.15665'
article_processing_charge: No
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Filippo
full_name: Quattrocchi, Filippo
id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
last_name: Quattrocchi
orcid: 0009-0000-9773-1931
citation:
ama: 'Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part
1: Second-order discrepancies between probability measures. arXiv. doi:10.48550/arXiv.2502.15665'
apa: 'Brigati, G., Maas, J., & Quattrocchi, F. (n.d.). Kinetic Optimal Transport
(OTIKIN) -- Part 1: Second-order discrepancies between probability measures. arXiv.
https://doi.org/10.48550/arXiv.2502.15665'
chicago: 'Brigati, Giovanni, Jan Maas, and Filippo Quattrocchi. “Kinetic Optimal
Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2502.15665.'
ieee: 'G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN)
-- Part 1: Second-order discrepancies between probability measures,” arXiv.
.'
ista: 'Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part
1: Second-order discrepancies between probability measures. arXiv, 2502.15665.'
mla: 'Brigati, Giovanni, et al. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order
Discrepancies between Probability Measures.” ArXiv, 2502.15665, doi:10.48550/arXiv.2502.15665.'
short: G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-28T13:12:08Z
date_published: 2025-08-10T00:00:00Z
date_updated: 2026-09-09T22:31:04Z
day: '10'
department:
- _id: GradSch
- _id: JaMa
doi: 10.48550/arXiv.2502.15665
ec_funded: 1
external_id:
arxiv:
- '2502.15665'
keyword:
- optimal transport
- kinetic theory
- second-order discrepancy
- Vlasov equation
- Wasserstein splines.
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2502.15665
month: '08'
oa: 1
oa_version: Preprint
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: arXiv
publication_status: draft
related_material:
record:
- id: '20563'
relation: dissertation_contains
status: public
status: public
title: 'Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between
probability measures'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '17277'
abstract:
- lang: eng
text: "This paper is devoted to stability results for the Gaussian logarithmic Sobolev
inequality, with explicit stability constants.\r\n\r\n"
acknowledgement: The authors thank Max Fathi and Pierre Cardaliaguet for fruitful
discussions and Emanuel Indrei for stimulating interactions. They also thank an
anonymous referee for useful comments and suggestions which have led to an improvement
of the manuscript. They also want to express their gratitude to the managing editor,
L. Gross, for his encouragements and questions. G.B. has been funded by the European
Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie
grant agreement No 754362. This work has been (partially) supported by the Project
Conviviality ANR-23-CE40-0003 of the French National Research Agency.
article_number: '110562'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Giovanni
full_name: Brigati, Giovanni
id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
last_name: Brigati
- first_name: Jean
full_name: Dolbeault, Jean
last_name: Dolbeault
- first_name: Nikita
full_name: Simonov, Nikita
last_name: Simonov
citation:
ama: Brigati G, Dolbeault J, Simonov N. Stability for the logarithmic Sobolev inequality.
Journal of Functional Analysis. 2024;287(8). doi:10.1016/j.jfa.2024.110562
apa: Brigati, G., Dolbeault, J., & Simonov, N. (2024). Stability for the logarithmic
Sobolev inequality. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2024.110562
chicago: Brigati, Giovanni, Jean Dolbeault, and Nikita Simonov. “Stability for the
Logarithmic Sobolev Inequality.” Journal of Functional Analysis. Elsevier,
2024. https://doi.org/10.1016/j.jfa.2024.110562.
ieee: G. Brigati, J. Dolbeault, and N. Simonov, “Stability for the logarithmic Sobolev
inequality,” Journal of Functional Analysis, vol. 287, no. 8. Elsevier,
2024.
ista: Brigati G, Dolbeault J, Simonov N. 2024. Stability for the logarithmic Sobolev
inequality. Journal of Functional Analysis. 287(8), 110562.
mla: Brigati, Giovanni, et al. “Stability for the Logarithmic Sobolev Inequality.”
Journal of Functional Analysis, vol. 287, no. 8, 110562, Elsevier, 2024,
doi:10.1016/j.jfa.2024.110562.
short: G. Brigati, J. Dolbeault, N. Simonov, Journal of Functional Analysis 287
(2024).
corr_author: '1'
date_created: 2024-07-21T22:01:00Z
date_published: 2024-10-15T00:00:00Z
date_updated: 2025-09-08T08:25:34Z
day: '15'
department:
- _id: JaMa
doi: 10.1016/j.jfa.2024.110562
external_id:
arxiv:
- '2303.12926'
isi:
- '001271814000001'
intvolume: ' 287'
isi: 1
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: 10.48550/arXiv.2303.12926
month: '10'
oa: 1
oa_version: Preprint
publication: Journal of Functional Analysis
publication_identifier:
eissn:
- 1096-0783
issn:
- 0022-1236
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Stability for the logarithmic Sobolev inequality
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 287
year: '2024'
...
---
OA_place: repository
_id: '17353'
abstract:
- lang: eng
text: "In this paper we derive estimates for the Hessian of the logarithm\r\n(log-Hessian)
for solutions to the heat equation. For initial data in the form\r\nof log-Lipschitz
perturbation of strongly log-concave measures, the log-Hessian\r\nadmits an explicit,
uniform (in space) lower bound. This yields a new estimate\r\nfor the Lipschitz
constant of a transport map pushing forward the standard\r\nGaussian to a measure
in this class. Further connections are discussed with\r\nscore-based diffusion
models and improved Gaussian logarithmic Sobolev\r\ninequalities. Finally, we
show that assuming only fast decay of the tails of\r\nthe initial datum does not
suffice to guarantee uniform log-Hessian upper\r\nbounds."
article_number: '2404.15205'
article_processing_charge: No
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.
arXiv. doi:10.48550/arXiv.2404.15205
apa: Brigati, G., & Pedrotti, F. (n.d.). Heat flow, log-concavity, and Lipschitz
transport maps. arXiv. https://doi.org/10.48550/arXiv.2404.15205
chicago: Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and
Lipschitz Transport Maps.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2404.15205.
ieee: G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport
maps,” arXiv. .
ista: Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps.
arXiv, 2404.15205.
mla: Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz
Transport Maps.” ArXiv, 2404.15205, doi:10.48550/arXiv.2404.15205.
short: G. Brigati, F. Pedrotti, ArXiv (n.d.).
corr_author: '1'
date_created: 2024-07-31T08:17:14Z
date_published: 2024-05-08T00:00:00Z
date_updated: 2026-04-07T13:00:02Z
day: '08'
department:
- _id: JaMa
doi: 10.48550/arXiv.2404.15205
external_id:
arxiv:
- '2404.15205'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2404.15205
month: '05'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '20591'
relation: later_version
status: public
- id: '17336'
relation: dissertation_contains
status: public
status: public
title: Heat flow, log-concavity, and Lipschitz transport maps
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...