Wasserstein metric and subordination

Philippe Clément ¹   ; Wolfgang Desch ²

¹ Mathematical Institute Leiden University P.O. Box 9512 NL-2300 RA Leiden, The Netherlands
² Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstrasse 36 8010 Graz, Austria

Studia Mathematica, Tome 189 (2008) no. 1, pp. 35-52

Résumé

Let $(X,d_X)$, $({\mit\Omega},d_{{\mit\Omega}})$ be complete separable metric spaces. Denote by $\mathcal P(X)$ the space of probability measures on $X$, by $W_p$ the $p$-Wasserstein metric with some $p \in [1,\infty)$, and by $\mathcal P_p(X)$ the space of probability measures on $X$ with finite Wasserstein distance from any point measure. Let $f: {\mit\Omega} \to \mathcal P_p(X)$, $\omega \mapsto f_{\omega}$, be a Borel map such that $f$ is a contraction from $({\mit\Omega},d_{{\mit\Omega}})$ into $(\mathcal P_p(X),W_p)$. Let $\nu_1,\nu_2$ be probability measures on ${\mit\Omega}$ with $W_p(\nu_1,\nu_2)$ finite. On $X$ we consider the subordinated measures $\mu_i=\int_{{\mit\Omega}}f_{\omega} \, d\nu_i(\omega).$ Then $W_p(\mu_1,\mu_2) \le W_p(\nu_1,\nu_2).$ As an application we show that the solution measures $\varrho_{\alpha}(t)$ to the partial differential equation \[ \frac{\partial}{\partial t}\varrho_{\alpha}(t) = -(-{\mit\Delta})^{\alpha/2}\varrho_{\alpha}(t), \quad \varrho_{\alpha}(0) = \delta_0 \quad \hbox{(the Dirac measure at 0)}, \] depend absolutely continuously on $t$ with respect to the Wasserstein metric $W_p$ whenever $1\le p \alpha 2$.

Détail

DOI : 10.4064/sm189-1-4

Mots-clés : mit omega mit omega complete separable metric spaces denote mathcal space probability measures p wasserstein metric infty mathcal space probability measures finite wasserstein distance point measure mit omega mathcal omega mapsto omega borel map contraction mit omega mit omega mathcal probability measures mit omega finite consider subordinated measures int mit omega omega omega application solution measures varrho alpha partial differential equation frac partial partial varrho alpha mit delta alpha varrho alpha quad varrho alpha delta quad hbox dirac measure depend absolutely continuously respect wasserstein metric whenever alpha