---
_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'
...