  Motivated both by theoretical and practical considerations in topological
data analysis, we generalize the $p$-Wasserstein distance on barcodes to
multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact
introduce two such generalizations $d_{\mathcal I}^p$ and $d_{\mathcal M}^p$,
such that $d_{\mathcal I}^\infty$ equals the interleaving distance and
$d_{\mathcal M}^\infty$ equals the matching distance. We show that on 1- or
2-parameter persistence modules over prime fields, $d_{\mathcal I}^p$ is the
universal (i.e., largest) metric satisfying a natural stability property; this
extends a stability theorem of Skraba and Turner for the $p$-Wasserstein
distance on barcodes in the 1-parameter case, and is also a close analogue of a
universality property for the interleaving distance given by the second author.
We also show that $d_{\mathcal M}^p\leq d_{\mathcal I}^p$ for all $p\in
[1,\infty]$, extending an observation of Landi in the $p=\infty$ case. We
observe that on 2-parameter persistence modules, $d_{\mathcal M}^p$ can be
efficiently approximated. In a forthcoming companion paper, we apply some of
these results to study the stability of ($2$-parameter) multicover persistent
homology.
