be given by $\langle\eta,e^{-in\theta}\rangle=a_n'(0)$. Since each $\mu_t$ is positive, the criterion must hold for each $t{\geqslant}0$. Thus the condition in Theorem \[thm:families\] is equivalent to requiring that the Fourier coefficients $a_n'(0)$ of $\eta$ satisfy $$\sum_m\sum_na'_{m-n}(0)\lambda_m\overline{\lambda_n}{\geqslant}0$$ whenever $\lambda_0,\lambda_1,\dots,\lambda_N$ are complex numbers such that $$\sum_m\sum_na_{m-n}(0)\lambda_m\overline{\lambda_n}=0.$$

Mass transportation {#sec:masstransport}
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#### Motivation.

For simplicity let $P={\mathbb{R}}^n$, $n{\geqslant}1$. Recall that for $p> 1$ the Wasserstein metric between two probabilities $\mu$ and $\nu$ on ${\mathbb{R}}^n$ with finite $p^\textrm{th}$ momenta on ${\mathbb{R}}^n$ can be defined as $$W_p(\mu,\nu)=\left(\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\operatorname{dist}(x,y)^p\,d\gamma(x,y)\right)^{\frac1p},$$ where the infimum is taken over all measures $\gamma$ on ${\mathbb{R}}^n\times {\mathbb{R}}^n$ whose marginals are $\mu$ and $\nu$. When $(\mu_t)_{t\in{\mathbb{R}}}$ is a family of smooth probability densities that defines an absolutely continuous curve in the space of probabilities endowed with the metric $W_p$, it has been shown [@ambrosiogiglisavare Chapter 8] that the derivative of $(\mu_t)_t$ exists for almost every $t$, and can be interpreted as the divergence of a vector field, that is, there are vector fields $v_t\colon {\mathbb{R}}^n\to{\mathbb{R}}^n$ that satisfy the *continuity equation*, $$\label{eq:continuityeq}
 \frac{d\mu_t}{dt}+\operatorname{div}(\mu_tv_t)=0,$$ for almost every $t\in{\mathbb{R}}$. The interpretation, which follows from Gauß’s theorem, is that the mass of $\mu_t$ is being carried or transported by the flow of the vector field $v_t$. This gives 