I want to calculate the distance between two probability measures $\mu$ and $\nu$ over the compact set $[a,b]\subset\mathbb R$, using the Wasserstein formula:
$$W=\inf_\varphi\int|x-y|\,{\rm d}\varphi(x,y),$$
where $\varphi$ is interpreted as a transport plan satisfying $\int\varphi(x,y)dy=\mu(x)$ and $\int\varphi(y,x)dy=\nu(x)$.
The particularity of my problem is that I know that $\nu$ is obtained from $\mu$ by moving mass to the right. More specifically, there exists a nondecreasing onto function $t:[a,b]\rightarrow[a,b]$ with $t(a)=a$ and $t(b)=b$ such that
$$ \nu(t(x))=\mu(x),\quad\text{for all $x\in[a,b]$.} $$
Then, can we say that $t$ corresponds to an optimal transport plan? That is, is the plan $$ \varphi^*(x,y)=\left\{ \begin{array}{ll} \mu(x)&\text{if $y=t(x)$}\\ 0&\text{otherwise} \end{array} \right. $$ optimal?
Thank you!