---
_id: '11739'
abstract:
- lang: eng
  text: We consider finite-volume approximations of Fokker--Planck equations on bounded
    convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures.
    We reprove the convergence of the discrete to continuous Fokker--Planck equation
    via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit
    at the level of the gradient flow structures, generalizing the one-dimensional
    result obtained by Disser and Liero. The proof is of variational nature and relies
    on a Mosco convergence result for functionals in the discrete-to-continuum limit
    that is of independent interest. Our results apply to arbitrary regular meshes,
    even though the associated discrete transport distances may fail to converge to
    the Wasserstein distance in this generality.
acknowledgement: This work was supported by the European Research Council (ERC) under
  the European Union's Horizon 2020 Research and Innovation Programme grant 716117
  and by the AustrianScience Fund (FWF) through grants F65 and W1245.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Dominik L
  full_name: Forkert, Dominik L
  id: 35C79D68-F248-11E8-B48F-1D18A9856A87
  last_name: Forkert
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Lorenzo
  full_name: Portinale, Lorenzo
  id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
  last_name: Portinale
citation:
  ama: Forkert DL, Maas J, Portinale L. Evolutionary $\Gamma$-convergence of entropic
    gradient flow structures for Fokker-Planck equations in multiple dimensions. <i>SIAM
    Journal on Mathematical Analysis</i>. 2022;54(4):4297-4333. doi:<a href="https://doi.org/10.1137/21M1410968">10.1137/21M1410968</a>
  apa: Forkert, D. L., Maas, J., &#38; Portinale, L. (2022). Evolutionary $\Gamma$-convergence
    of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied
    Mathematics. <a href="https://doi.org/10.1137/21M1410968">https://doi.org/10.1137/21M1410968</a>
  chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\Gamma$-Convergence
    of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
    <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied
    Mathematics, 2022. <a href="https://doi.org/10.1137/21M1410968">https://doi.org/10.1137/21M1410968</a>.
  ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\Gamma$-convergence
    of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,”
    <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4. Society for Industrial
    and Applied Mathematics, pp. 4297–4333, 2022.
  ista: Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\Gamma$-convergence of
    entropic gradient flow structures for Fokker-Planck equations in multiple dimensions.
    SIAM Journal on Mathematical Analysis. 54(4), 4297–4333.
  mla: Forkert, Dominik L., et al. “Evolutionary $\Gamma$-Convergence of Entropic
    Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.”
    <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 4, Society for Industrial
    and Applied Mathematics, 2022, pp. 4297–333, doi:<a href="https://doi.org/10.1137/21M1410968">10.1137/21M1410968</a>.
  short: D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis
    54 (2022) 4297–4333.
date_created: 2022-08-07T22:01:59Z
date_published: 2022-07-18T00:00:00Z
date_updated: 2023-08-03T12:37:21Z
day: '18'
department:
- _id: JaMa
doi: 10.1137/21M1410968
ec_funded: 1
external_id:
  arxiv:
  - '2008.10962'
  isi:
  - '000889274600001'
intvolume: '        54'
isi: 1
issue: '4'
keyword:
- Fokker--Planck equation
- gradient flow
- evolutionary $\Gamma$-convergence
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2008.10962'
month: '07'
oa: 1
oa_version: Preprint
page: 4297-4333
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
- _id: 260788DE-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
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'
related_material:
  record:
  - id: '10022'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Evolutionary $\Gamma$-convergence of entropic gradient flow structures for
  Fokker-Planck equations in multiple dimensions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...
---
_id: '10011'
abstract:
- lang: eng
  text: We propose a new weak solution concept for (two-phase) mean curvature flow
    which enjoys both (unconditional) existence and (weak-strong) uniqueness properties.
    These solutions are evolving varifolds, just as in Brakke's formulation, but are
    coupled to the phase volumes by a simple transport equation. First, we show that,
    in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461,
    (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold
    solution in our sense. Second, we prove that any calibrated flow in the sense
    of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean
    curvature flow - is unique in the class of our new varifold solutions. This is
    in sharp contrast to the case of Brakke flows, which a priori may disappear at
    any given time and are therefore fatally non-unique. Finally, we propose an extension
    of the solution concept to the multi-phase case which is at least guaranteed to
    satisfy a weak-strong uniqueness principle.
acknowledgement: This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG,
  German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.
  The content of this paper was developed and parts of it were written during a visit
  of the first author to the Hausdorff Center of Mathematics (HCM), University of
  Bonn. The hospitality and the support of HCM are gratefully acknowledged.
article_number: '2109.04233'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastian
  full_name: Hensel, Sebastian
  id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
  last_name: Hensel
  orcid: 0000-0001-7252-8072
- first_name: Tim
  full_name: Laux, Tim
  last_name: Laux
citation:
  ama: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
    Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.2109.04233">10.48550/arXiv.2109.04233</a>'
  apa: 'Hensel, S., &#38; Laux, T. (n.d.). A new varifold solution concept for mean
    curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2109.04233">https://doi.org/10.48550/arXiv.2109.04233</a>'
  chicago: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for
    Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.”
    <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2109.04233">https://doi.org/10.48550/arXiv.2109.04233</a>.'
  ieee: 'S. Hensel and T. Laux, “A new varifold solution concept for mean curvature
    flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness,” <i>arXiv</i>.
    .'
  ista: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
    Convergence of  the Allen-Cahn equation and weak-strong uniqueness. arXiv, 2109.04233.'
  mla: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean
    Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.”
    <i>ArXiv</i>, 2109.04233, doi:<a href="https://doi.org/10.48550/arXiv.2109.04233">10.48550/arXiv.2109.04233</a>.'
  short: S. Hensel, T. Laux, ArXiv (n.d.).
date_created: 2021-09-13T12:17:10Z
date_published: 2021-09-09T00:00:00Z
date_updated: 2023-05-03T10:34:38Z
day: '09'
department:
- _id: JuFi
doi: 10.48550/arXiv.2109.04233
ec_funded: 1
external_id:
  arxiv:
  - '2109.04233'
keyword:
- Mean curvature flow
- gradient flows
- varifolds
- weak solutions
- weak-strong uniqueness
- calibrated geometry
- gradient-flow calibrations
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2109.04233
month: '09'
oa: 1
oa_version: Preprint
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
  call_identifier: H2020
  grant_number: '948819'
  name: Bridging Scales in Random Materials
publication: arXiv
publication_status: submitted
status: public
title: 'A new varifold solution concept for mean curvature flow: Convergence of  the
  Allen-Cahn equation and weak-strong uniqueness'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '10023'
abstract:
- lang: eng
  text: We study the temporal dissipation of variance and relative entropy for ergodic
    Markov Chains in continuous time, and compute explicitly the corresponding dissipation
    rates. These are identified, as is well known, in the case of the variance in
    terms of an appropriate Hilbertian norm; and in the case of the relative entropy,
    in terms of a Dirichlet form which morphs into a version of the familiar Fisher
    information under conditions of detailed balance. Here we obtain trajectorial
    versions of these results, valid along almost every path of the random motion
    and most transparent in the backwards direction of time. Martingale arguments
    and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer
    and Tschiderer for conservative diffusions. Extensions are developed to general
    “convex divergences” and to countable state-spaces. The steepest descent and gradient
    flow properties for the variance, the relative entropy, and appropriate generalizations,
    are studied along with their respective geometries under conditions of detailed
    balance, leading to a very direct proof for the HWI inequality of Otto and Villani
    in the present context.
acknowledgement: I.K. acknowledges support from the U.S. National Science Foundation
  under Grant NSF-DMS-20-04997. J.M. acknowledges support from the European Research
  Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (grant agreement No 716117) and from the Austrian Science Fund (FWF) through project
  F65. W.S. acknowledges support from the Austrian Science Fund (FWF) under grant
  P28861 and by the Vienna Science and Technology Fund (WWTF) through projects MA14-008
  and MA16-021.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Ioannis
  full_name: Karatzas, Ioannis
  last_name: Karatzas
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Walter
  full_name: Schachermayer, Walter
  last_name: Schachermayer
citation:
  ama: Karatzas I, Maas J, Schachermayer W. Trajectorial dissipation and gradient
    flow for the relative entropy in Markov chains. <i>Communications in Information
    and Systems</i>. 2021;21(4):481-536. doi:<a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">10.4310/CIS.2021.v21.n4.a1</a>
  apa: Karatzas, I., Maas, J., &#38; Schachermayer, W. (2021). Trajectorial dissipation
    and gradient flow for the relative entropy in Markov chains. <i>Communications
    in Information and Systems</i>. International Press. <a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a>
  chicago: Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation
    and Gradient Flow for the Relative Entropy in Markov Chains.” <i>Communications
    in Information and Systems</i>. International Press, 2021. <a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a>.
  ieee: I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and
    gradient flow for the relative entropy in Markov chains,” <i>Communications in
    Information and Systems</i>, vol. 21, no. 4. International Press, pp. 481–536,
    2021.
  ista: Karatzas I, Maas J, Schachermayer W. 2021. Trajectorial dissipation and gradient
    flow for the relative entropy in Markov chains. Communications in Information
    and Systems. 21(4), 481–536.
  mla: Karatzas, Ioannis, et al. “Trajectorial Dissipation and Gradient Flow for the
    Relative Entropy in Markov Chains.” <i>Communications in Information and Systems</i>,
    vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:<a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">10.4310/CIS.2021.v21.n4.a1</a>.
  short: I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and
    Systems 21 (2021) 481–536.
date_created: 2021-09-19T08:53:19Z
date_published: 2021-06-04T00:00:00Z
date_updated: 2021-09-20T12:51:18Z
day: '04'
department:
- _id: JaMa
doi: 10.4310/CIS.2021.v21.n4.a1
ec_funded: 1
external_id:
  arxiv:
  - '2005.14177'
intvolume: '        21'
issue: '4'
keyword:
- Markov Chain
- relative entropy
- time reversal
- steepest descent
- gradient flow
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2005.14177
month: '06'
oa: 1
oa_version: Preprint
page: 481-536
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: Communications in Information and Systems
publication_identifier:
  issn:
  - 1526-7555
publication_status: published
publisher: International Press
quality_controlled: '1'
status: public
title: Trajectorial dissipation and gradient flow for the relative entropy in Markov
  chains
type: journal_article
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 21
year: '2021'
...