Gradient flows of the entropy for jump processes

Erbar, Matthias

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 920-945.

On considère une nouvelle distance entre les mesures de probabilité sur ${ℝ}^n$. 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.

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.

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