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

Charles Boubel (1) , Nicolas Juillet (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let µ = (µt) t∈R be a 1-parameter family of probability measures on R. In [13] we introduced its "Markov-quantile" process: a process X = (Xt) t∈R that resembles at most the quantile process attached to µ, among the Markov processes attached 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 P 2 (R). Then, X is solution of a Benamou-Brenier transport problem with intermediate marginals µt. It provides a Markov Lagrangian probabilistic representation of the continuity equation, moreover the unique Markov process:-obtained as a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times.-or, alternatively, obtained as a limit of processes linearly interpolating µ. This raises new questions about ways to obtain Markov Lagrangian representations of the continuity equation, and to seek uniqueness properties in this framework.

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Charles Boubel, Nicolas Juillet. On absolutely continuous curves in the Wasserstein space over R and their representation by an optimal Markov process. 2021. ⟨hal-03218874⟩