Fractional Sobolev paths on Wasserstein spaces and their energy-minimizing particle representations

TL;DR

This paper explores the existence of energy-minimizing stochastic processes with prescribed time-dependent distributions in Wasserstein spaces, focusing on fractional Sobolev regularity and deterministic paths.

Contribution

It establishes conditions for the existence of energy-minimizing processes with fractional Sobolev regularity on Wasserstein spaces in a deterministic setting.

Findings

Proved existence of energy-minimizing processes under fractional Sobolev regularity conditions.

Provided conditions for processes to realize the regularity of given paths.

Linked deterministic paths in Wasserstein spaces to solutions of the continuity equation.

Abstract

We study a generalization of the Monge--Kantorovich optimal transport problem. Given a prescribed family of time-dependent probability measures $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide with $(\mu_t)$ (if there is any), a process that minimizes a given energy. After discussing a sufficient condition for the energy to ensure the existence of a minimizer, we investigate fractional Sobolev energies. Given a deterministic path $(\mu_t)$ on a $p$-Wasserstein space with fractional Sobolev regularity $W^{\alpha,p}$, where $1/p < \alpha < 1$, we provide conditions under which we prove the existence of a process that minimizes the energy and construct a process that realizes the regularity of $(\mu_t)$. While continuous paths of low regularity on Wasserstein spaces naturally appear in stochastic analysis, they can also arise deterministically as solutions to the continuity equation. This paper is devoted to the deterministic setting to gain some understanding of the required conditions. The subsequent companion paper (arXiv:2503.10859) focuses on the stochastic setting and applications to SPDEs.