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Displacement convexity for the entropy in semi-discrete non-linear Fokker–Planck equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Global differential geometry Markov processes

Published online by Cambridge University Press:  10 January 2018

JOSÉ A. CARRILLO ,
ANSGAR JÜNGEL  and
MATHEUS C. SANTOS
JOSÉ A. CARRILLO
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: carrillo@imperial.ac.uk
ANSGAR JÜNGEL
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria email: juengel@tuwien.ac.at
MATHEUS C. SANTOS
Affiliation:
Departamento de Matemática – IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil email: matheus.santos@ufrgs.com
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Abstract

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The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

Keywords

Entropydisplacement convexitylogarithmic meanfinite differencesfast-diffusion equation

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 65M20: Method of lines
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1103 - 1122
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge University Press 2018

Footnotes

†

The first author was partially supported by the Royal Society via a Wolfson Research Merit Award and the EPSRC Grant EP/P031587/1. The second author acknowledges partial support from the Austrian Science Fund (FWF), Grants P22108, P24304, F65 and W1245. The last author acknowledges the support from the São Paulo Research Foundation (FAPESP), Grant 2015/20962-7.

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