On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process
Charles Boubel, Nicolas Juillet
TL;DR
This work constructs a Markov, minimal Lagrangian probabilistic representation for absolutely continuous curves in the Wasserstein space $\mathcal{P}_2(\mathbb{R})$ by introducing the Markov-quantile process $\mathfrak{MQ}$ and proving $\mathcal{A}(\mathfrak{MQ})=\mathcal{E}(\mu)$ for curves with finite energy $\mathcal{E}(\mu)$. The main results show that $\mathfrak{MQ}$ arises as a Markov limit of quantile-based discretizations $\mathfrak{Q}_{[R_n]}$, and, under natural conditions, is the unique Markov minimal Lagrangian representative within a certain class; a second construction via dispersion-interpolating processes yields a Markov limit in the same spirit, at least in dimension one. The paper also discusses non-uniqueness phenomena among Markov minimal representatives and outlines open questions for higher dimensions and alternative constructions, connecting to broader themes in dynamical optimal transport and Kellerer-type problems. Overall, it advances a probabilistic, Markovian viewpoint on dynamical transport with prescribed marginals, with potential implications for higher-dimensional and Schrödinger-regularized transport problems.
Abstract
Let $μ$ = ($μ$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 $μ$, among the Markov processesattached to $μ$, i.e. whose family of marginal laws is $μ$.In this article we look at the case where $μ$ is absolutely continuous in the Wasserstein space P2(R). Then X is solution of adynamical transport problem with marginals ($μ$t)t. It provides a Markov minimal Lagrangian probabilistic representative of $μ$, 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 $μ$.This raises new questions about ways to obtain Markov Lagrangian representatives, and to seek uniqueness properties in thisframework.