respectively. We shall consider ${\mathcal{P}}_p({\mathbb{R}})$ as a metric space, equipped with the $L^p$-Wasserstein distance $W_p$, see Lemma \[lem:17\] below for a definition. By abuse of notation, we will frequently identify absolutely continuous measures with their density functions. Two concepts of (weak) convergence on ${\mathfrak{M}_+}({\mathbb{R}})$ will be used: $\mu_n\to\mu$ *narrowly* means that $$\begin{aligned}
  \label{eq:narrow}
  \int_{\mathbb{R}}\varphi(x){\, \mathrm{d}}\mu_n(x)\to\int_{\mathbb{R}}\varphi(x){\, \mathrm{d}}\mu(x)  \end{aligned}$$ holds for every bounded $\varphi\in C^0({\mathbb{R}})$, and $\mu_n\to\mu$ *vaguely* means that is true for all $\varphi\in C^0({\mathbb{R}})$ of compact support. Note that $\mu_n({\mathbb{R}})\to\mu({\mathbb{R}})$ if $\mu_n\to\mu$ narrowly, but not necessarily if $\mu_n\to\mu$ only vaguely.

Lebesgue spaces are denoted by $L^p({\mathbb{R}})$, with $p\in[1,\infty]$, and $L^\infty_c({\mathbb{R}})$ is the space of essentially bounded functions with compact support. Finally, we denote by ${\mathrm{D}}(I)$ the linear space of *càdlàg functions* over the interval $I\subseteq{\mathbb{R}}$, where either $I={\mathbb{R}}$ or $I=[a,b)$. That is, ${\mathrm{D}}(I)$ is the space of bounded functions $X:I\to{\mathbb{R}}$, which are right continuous and have a left limit at every point. We endow ${\mathrm{D}}(I)$ with the $\sup$-norm (*not* the Skorokhod topology), which makes it a Banach space.

The Pseudo-inverse {#sec:pseudo-inverse-techn}
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Definition and elementary properties {#sec:pseudo-inverse-elem}
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In one spatial dimension, we can exploit a special transformation technique which makes equation much more amenable to estimates in the Wasserstein 