Estimates on Path Functionals over Wasserstein Spaces

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] $\mathcal{G}_{r,p}$ defined on Lipschitz curves $\gamma$ valued in the p-Wasserstein space. The problem considered is the following: given a measure $\mu$, give conditions in order to assure the existence of a curve $\gamma$ such that $\gamma(0)=\mu$, $\gamma(1)=\delta_{x_0}$, and $\mathcal{G}_{r,p}(\gamma)<+\infty$. To this end, new estimates on $\mathcal{G}_{r,p}(\mu)$ 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.