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On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

Emmanuel Boissard, Thibaut Le Gouic (2014)

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

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably...

On the motion of a curve by its binormal curvature

Jerrard, Robert L., Didier Smets (2015)

Journal of the European Mathematical Society

We propose a weak formulation for the binormal curvature flow of curves in 3 . This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

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