Wasserstein metric and subordination
Volume 189 / 2008
Abstract
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 $1\le p < \alpha < 2$.
