Gradient flows of the entropy for jump processes

Matthias Erbar

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 920-945
  • ISSN: 0246-0203

Abstract

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We introduce a new transport distance between probability measures on ℝ d that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is convex along geodesics.

How to cite

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Erbar, Matthias. "Gradient flows of the entropy for jump processes." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 920-945. <http://eudml.org/doc/272002>.

@article{Erbar2014,
abstract = {We introduce a new transport distance between probability measures on $\mathbb \{R\}^\{d\}$ that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is convex along geodesics.},
author = {Erbar, Matthias},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {jump process; Lévy process; gradient flow; entropy; optimal transport; jump processes; Lévy processes; non-local operator},
language = {eng},
number = {3},
pages = {920-945},
publisher = {Gauthier-Villars},
title = {Gradient flows of the entropy for jump processes},
url = {http://eudml.org/doc/272002},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Erbar, Matthias
TI - Gradient flows of the entropy for jump processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 920
EP - 945
AB - We introduce a new transport distance between probability measures on $\mathbb {R}^{d}$ that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is convex along geodesics.
LA - eng
KW - jump process; Lévy process; gradient flow; entropy; optimal transport; jump processes; Lévy processes; non-local operator
UR - http://eudml.org/doc/272002
ER -

References

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