# Path Functionals over Wasserstein Spaces

### Giuseppe Buttazzo

Università di Pisa, Italy### Alessio Brancolini

Politecnico di Bari, Italy### Filippo Santambrogio

Scuola Normale Superiore, Pisa, Italy

## Abstract

Given a metric space $X$ we consider a general class of functionals which measure the cost of a path in $X$ joining two given points $x_0$ and $x_1$, providing abstract existence results for optimal paths. The results are then applied to the case when $X$ is a Wasserstein space of probabilities on a given set $\Omega$ and the cost of a path depends on the value of classical functionals over measures. Conditions to link arbitrary extremal measures $\mu_0$ and $\mu_1$ by means of finite cost paths are given.

## Cite this article

Giuseppe Buttazzo, Alessio Brancolini, Filippo Santambrogio, Path Functionals over Wasserstein Spaces. J. Eur. Math. Soc. 8 (2006), no. 3, pp. 415–434

DOI 10.4171/JEMS/61