On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process

TL;DR

This paper studies absolutely continuous curves in the Wasserstein space over R, introducing a unique Markov process called the Markov-quantile process that minimally represents these curves and solves a dynamic transport problem.

Contribution

It introduces the Markov-quantile process as a unique Markovian Lagrangian representative for absolutely continuous Wasserstein curves, linking optimal transport and Markov processes.

Findings

The Markov-quantile process is the unique Markovian solution for absolutely continuous Wasserstein curves.

It provides a minimal Lagrangian probabilistic representation of the curve.

The process is characterized as a limit of quantile processes with specific independence and interpolation properties.

Abstract

Let $\mu$ = ($\mu$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $\mu$, among the Markov processesattached to $\mu$, i.e. whose family of marginal laws is $\mu$.In this article we look at the case where $\mu$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($\mu$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $\mu$, whichis moreover unique among the processes obtained as certain types of limits: limits for the finite dimensional topology of quantileprocesses where the past is made independent of the future conditionally on the present at finitely many times, or limits of processeslinearly interpolating $\mu$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework.