---
_id: '7629'
abstract:
- lang: eng
text: "This thesis is based on three main topics: In the first part, we study convergence
of discrete gradient flow structures associated with regular finite-volume discretisations
of Fokker-Planck equations. We show evolutionary I convergence of the discrete
gradient flows to the L2-Wasserstein gradient flow corresponding to the solution
of a Fokker-Planck\r\nequation in arbitrary dimension d >= 1. Along the argument,
we prove Mosco- and I-convergence results for discrete energy functionals, which
are of independent interest for convergence of equivalent gradient flow structures
in Hilbert spaces.\r\nThe second part investigates L2-Wasserstein flows on metric
graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein
distance, which is proved via a regularisation scheme for solutions of the continuity
equation, adapted to the peculiar geometric structure of metric graphs. Based
on those results, we show that the L2-Wasserstein space over a metric graph admits
a gradient flow which may be identified as a solution of a Fokker-Planck equation.\r\nIn
the third part, we focus again on the discrete gradient flows, already encountered
in the first part. We propose a variational structure which extends the gradient
flow structure to Markov chains violating the detailed-balance conditions. Using
this structure, we characterise contraction estimates for the discrete heat flow
in terms of convexity of\r\ncorresponding path-dependent energy functionals. In
addition, we use this approach to derive several functional inequalities for said
functionals."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Dominik L
full_name: Forkert, Dominik L
id: 35C79D68-F248-11E8-B48F-1D18A9856A87
last_name: Forkert
citation:
ama: Forkert DL. Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains. 2020. doi:10.15479/AT:ISTA:7629
apa: Forkert, D. L. (2020). Gradient flows in spaces of probability measures
for finite-volume schemes, metric graphs and non-reversible Markov chains.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:7629
chicago: Forkert, Dominik L. “Gradient Flows in Spaces of Probability Measures for
Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains.” Institute
of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:7629.
ieee: D. L. Forkert, “Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains,” Institute of Science
and Technology Austria, 2020.
ista: Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume
schemes, metric graphs and non-reversible Markov chains. Institute of Science
and Technology Austria.
mla: Forkert, Dominik L. Gradient Flows in Spaces of Probability Measures for
Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains. Institute
of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:7629.
short: D.L. Forkert, Gradient Flows in Spaces of Probability Measures for Finite-Volume
Schemes, Metric Graphs and Non-Reversible Markov Chains, Institute of Science
and Technology Austria, 2020.
date_created: 2020-04-02T06:40:23Z
date_published: 2020-03-31T00:00:00Z
date_updated: 2023-09-07T13:03:12Z
day: '31'
ddc:
- '510'
degree_awarded: PhD
department:
- _id: JaMa
doi: 10.15479/AT:ISTA:7629
ec_funded: 1
file:
- access_level: open_access
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creator: dernst
date_created: 2020-04-14T10:47:59Z
date_updated: 2020-07-14T12:48:01Z
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file_size: 1063908
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file_date_updated: 2020-07-14T12:48:01Z
has_accepted_license: '1'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: '154'
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
status: public
supervisor:
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
title: Gradient flows in spaces of probability measures for finite-volume schemes,
metric graphs and non-reversible Markov chains
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2020'
...