---
_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. Communications in Information
and Systems. 2021;21(4):481-536. doi:10.4310/CIS.2021.v21.n4.a1
apa: Karatzas, I., Maas, J., & Schachermayer, W. (2021). Trajectorial dissipation
and gradient flow for the relative entropy in Markov chains. Communications
in Information and Systems. International Press. https://doi.org/10.4310/CIS.2021.v21.n4.a1
chicago: Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation
and Gradient Flow for the Relative Entropy in Markov Chains.” Communications
in Information and Systems. International Press, 2021. https://doi.org/10.4310/CIS.2021.v21.n4.a1.
ieee: I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and
gradient flow for the relative entropy in Markov chains,” Communications in
Information and Systems, 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.” Communications in Information and Systems,
vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:10.4310/CIS.2021.v21.n4.a1.
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: 2025-04-14T07:27:45Z
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'
...
---
_id: '5587'
abstract:
- lang: eng
text: "Supporting material to the article \r\nSTATISTICAL MECHANICS FOR METABOLIC
NETWORKS IN STEADY-STATE GROWTH\r\n\r\nboundscoli.dat\r\nFlux Bounds of the E.
coli catabolic core model iAF1260 in a glucose limited minimal medium. \r\n\r\npolcoli.dat\r\nMatrix
enconding the polytope of the E. coli catabolic core model iAF1260 in a glucose
limited minimal medium, \r\nobtained from the soichiometric matrix by standard
linear algebra (reduced row echelon form).\r\n\r\nellis.dat\r\nApproximate Lowner-John
ellipsoid rounding the polytope of the E. coli catabolic core model iAF1260 in
a glucose limited minimal medium\r\nobtained with the Lovasz method.\r\n\r\npoint0.dat\r\nCenter
of the approximate Lowner-John ellipsoid rounding the polytope of the E. coli
catabolic core model iAF1260 in a glucose limited minimal medium\r\nobtained with
the Lovasz method.\r\n\r\nlovasz.cpp \r\nThis c++ code file receives in input
the polytope of the feasible steady states of a metabolic network, \r\n(matrix
and bounds), and it gives in output an approximate Lowner-John ellipsoid rounding
the polytope\r\nwith the Lovasz method \r\nNB inputs are referred by defaults
to the catabolic core of the E.Coli network iAF1260. \r\nFor further details we
refer to PLoS ONE 10.4 e0122670 (2015).\r\n\r\nsampleHRnew.cpp \r\nThis c++
code file receives in input the polytope of the feasible steady states of a metabolic
network, \r\n(matrix and bounds), the ellipsoid rounding the polytope, a point
inside and \r\nit gives in output a max entropy sampling at fixed average growth
rate \r\nof the steady states by performing an Hit-and-Run Monte Carlo Markov
chain.\r\nNB inputs are referred by defaults to the catabolic core of the E.Coli
network iAF1260. \r\nFor further details we refer to PLoS ONE 10.4 e0122670 (2015)."
article_processing_charge: No
author:
- first_name: Daniele
full_name: De Martino, Daniele
id: 3FF5848A-F248-11E8-B48F-1D18A9856A87
last_name: De Martino
orcid: 0000-0002-5214-4706
- first_name: Gasper
full_name: Tkacik, Gasper
id: 3D494DCA-F248-11E8-B48F-1D18A9856A87
last_name: Tkacik
orcid: 0000-0002-6699-1455
citation:
ama: De Martino D, Tkačik G. Supporting materials “STATISTICAL MECHANICS FOR METABOLIC
NETWORKS IN STEADY-STATE GROWTH.” 2018. doi:10.15479/AT:ISTA:62
apa: De Martino, D., & Tkačik, G. (2018). Supporting materials “STATISTICAL
MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.” Institute of Science
and Technology Austria. https://doi.org/10.15479/AT:ISTA:62
chicago: De Martino, Daniele, and Gašper Tkačik. “Supporting Materials ‘STATISTICAL
MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.’” Institute of Science
and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:62.
ieee: D. De Martino and G. Tkačik, “Supporting materials ‘STATISTICAL MECHANICS
FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.’” Institute of Science and Technology
Austria, 2018.
ista: De Martino D, Tkačik G. 2018. Supporting materials ‘STATISTICAL MECHANICS
FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH’, Institute of Science and Technology
Austria, 10.15479/AT:ISTA:62.
mla: De Martino, Daniele, and Gašper Tkačik. Supporting Materials “STATISTICAL
MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH.” Institute of Science
and Technology Austria, 2018, doi:10.15479/AT:ISTA:62.
short: D. De Martino, G. Tkačik, (2018).
datarep_id: '111'
date_created: 2018-12-12T12:31:41Z
date_published: 2018-09-21T00:00:00Z
date_updated: 2025-04-15T06:50:08Z
day: '21'
ddc:
- '530'
department:
- _id: GaTk
doi: 10.15479/AT:ISTA:62
ec_funded: 1
file:
- access_level: open_access
checksum: 97992e3e8cf8544ec985a48971708726
content_type: application/zip
creator: system
date_created: 2018-12-12T13:05:13Z
date_updated: 2020-07-14T12:47:08Z
file_id: '5641'
file_name: IST-2018-111-v1+1_CODES.zip
file_size: 14376
relation: main_file
file_date_updated: 2020-07-14T12:47:08Z
has_accepted_license: '1'
keyword:
- metabolic networks
- e.coli core
- maximum entropy
- monte carlo markov chain sampling
- ellipsoidal rounding
license: https://creativecommons.org/publicdomain/zero/1.0/
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 254E9036-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P28844-B27
name: Biophysics of information processing in gene regulation
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '161'
relation: research_paper
status: public
status: public
title: Supporting materials "STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE
GROWTH"
tmp:
image: /images/cc_0.png
legal_code_url: https://creativecommons.org/publicdomain/zero/1.0/legalcode
name: Creative Commons Public Domain Dedication (CC0 1.0)
short: CC0 (1.0)
type: research_data
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2018'
...