Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Gradient flows of the entropy for jump processes

Matthias Erbar

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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.


On considère une nouvelle distance entre les mesures de probabilité sur $\mathbb{R}^{n}$. Elle est construite à partir d’un processus de saut par une variante non-locale de la formule de Benamou–Brenier. Pour les processus de Lévy on démontre que le semigroupe engendré par l’opérateur non-local associé est le flot de gradient de l’entropie par rapport à la nouvelle distance. On démontre aussi que l’entropie est convexe le long des géodésiques dans ce cas.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 920-945.

First available in Project Euclid: 20 June 2014

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Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 35S10: Initial value problems for pseudodifferential operators 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 49J45: Methods involving semicontinuity and convergence; relaxation 60G51: Processes with independent increments; Lévy processes

Jump process Lévy process Gradient flow Entropy Optimal transport


Erbar, Matthias. Gradient flows of the entropy for jump processes. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 920--945. doi:10.1214/12-AIHP537.

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