Absolutely continuous and BV-curves in 1-Wasserstein spaces

TL;DR

This paper extends the superposition principle to 1-Wasserstein spaces by introducing BV-curves, allowing representation of curves with bounded variation and characterizing geodesics and the continuity equation.

Contribution

It generalizes the superposition principle to BV-curves in 1-Wasserstein spaces, addressing the lack of lifts for absolutely continuous curves in this setting.

Findings

BV-curves can be represented by probability measures encoding total variation

The result characterizes metric speed for BV-curves in 1-Wasserstein spaces

Applications include geodesic characterization and continuity equation analysis

Abstract

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting.