Estimates on path functionals over Wasserstein spaces

In this paper we consider the class a functionals (introduced by Brancolini, Buttazzo, and Santambrogio) Gr,p(γ) 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 a curve γ such that γ(0)=μ, γ(1)=δx0, and Gr,p(γ)<+∞. To this end, new estimates on Gr,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.