Gradient flows of the entropy for jump processes

By Matthias Erbar. 2012

67 citations recorded by the source corpus.

We introduce a new transportation distance between probability measures that is built from a L\'evy 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/726b9f6cca2f7ed3e6e93344b884bf556863a54e

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In Applied Mathematics

Citations in the corpus

Gradient flow structures for discrete porous medium equations (2012, background): “Related gradient flow structures have been found for fractional heat equations [7] and quantum mechanical evolution equations [4, 15].”
Displacement convexity of entropy and related inequalities on graphs (2012, background): “This is further developed by Erbar and Maas [13] who prove, inter alia, that if {νt }t∈[0,1] is a geodesic from ν0 to ν1, then H(νt |μ) ≤ (1 − t)H(ν0|μ) + t H(ν1|μ) − 1 n t (1 − t)W22 (ν0, ν1), ∀t ∈ [0, 1], where μ ≡ 1/2n is the uniform measure.”
On the difference between entropic cost and the optimal transport cost (2019, unknown): “See articles [20, 10, 8, 11, 6, 9].”