Title | Estimates on path functionals over Wasserstein Spaces |

Publication Type | Journal Article |

Year of Publication | 2010 |

Authors | Bianchini, S, Brancolini, A |

Journal | SIAM J. Math. Anal. 42 (2010) 1179-1217 |

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] $\\\\mathcal{G}_{r,p}$ 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. |

URL | http://hdl.handle.net/1963/3583 |

DOI | 10.1137/100782693 |

## Estimates on path functionals over Wasserstein Spaces

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