Estimates on path functionals over Wasserstein spaces

Bianchini S., Brancolini A.

Research article (journal) | Peer reviewed

Abstract

In this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] Qr,p defined on Lipschitz curves γ valued in the p-Wasserstein space. The problem considered is the following: given a measure μ, give conditions in order to assure the existence of a curve γ such that γ(0) = μ, γ(1) = δxo, and Qr, p(γ) < +00. To this end, new estimates on Qr, p(μ) are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters (r, p) for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions. © 2010 Societ y for Industrial and Applied Mathematics.