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The heat equation on manifolds as a gradient flow in the Wasserstein space

Matthias Erbar — 2010

Annales de l'I.H.P. Probabilités et statistiques

We study the gradient flow for the relative entropy functional on probability measures over a riemannian manifold. To this aim we present a notion of a riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.

Gradient flows of the entropy for jump processes

Matthias Erbar — 2014

Annales de l'I.H.P. Probabilités et statistiques

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...

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