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

Ehsan Abedi

TL;DR

The paper extends the Monge–Kantorovich framework to time-dependent marginals by optimizing over path measures (lifts) and introducing fractional Sobolev and Besov energies on continuous path spaces. It proves existence of energy-minimizing lifts under lower semicontinuity and compactness assumptions, and constructs realizing lifts when the time-marginal curve is compatible in the Wasserstein sense. A dynamic formulation of the Wasserstein distance is established via Besov energy, generalizing Benamou–Brenier to the fractional setting, with a superposition principle linking curve regularity to path measures. The work also clarifies the necessity of compatibility through counterexamples, and highlights the deterministic versus stochastic perspectives, with a companion paper addressing SPDE applications.

Abstract

We study a generalization of the Monge--Kantorovich optimal transport problem. Given a prescribed family of time-dependent probability measures $(μ_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide with $(μ_t)$ (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 $(μ_t)$ on a $p$-Wasserstein space with fractional Sobolev regularity $W^{α,p}$, where $1/p < α< 1$, 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 $(μ_t)$. 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.

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

TL;DR

The paper extends the Monge–Kantorovich framework to time-dependent marginals by optimizing over path measures (lifts) and introducing fractional Sobolev and Besov energies on continuous path spaces. It proves existence of energy-minimizing lifts under lower semicontinuity and compactness assumptions, and constructs realizing lifts when the time-marginal curve is compatible in the Wasserstein sense. A dynamic formulation of the Wasserstein distance is established via Besov energy, generalizing Benamou–Brenier to the fractional setting, with a superposition principle linking curve regularity to path measures. The work also clarifies the necessity of compatibility through counterexamples, and highlights the deterministic versus stochastic perspectives, with a companion paper addressing SPDE applications.

Abstract

We study a generalization of the Monge--Kantorovich optimal transport problem. Given a prescribed family of time-dependent probability measures , we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide with (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 on a -Wasserstein space with fractional Sobolev regularity , where , 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 . 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.

Paper Structure

This paper contains 27 sections, 45 theorems, 227 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Problem formulation and main results
  3. Motivating example
  4. Organization of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Continuous paths on metric spaces
  8. Hölder paths on metric spaces
  9. p-variation paths on metric spaces
  10. Fractional Sobolev paths on metric spaces
  11. Sobolev paths on metric spaces
  12. Geodesics on metric spaces
  13. Piecewise geodesic approximation on geodesic spaces
  14. Path measures and sample path regularity
  15. Probability measures, narrow convergence, and tightness
  16. ...and 12 more sections

Key Result

Proposition 1.1

Let $(\mathcal{X}, d)$ be a complete separable metric space, and $I \coloneqq [0,T] \subset \mathbb{R}$. Let $\Psi: C(I;\mathcal{X}) \to [0,+\infty]$ be ① a lower semi-continuous map ② whose sublevels are relatively compact in $C(I;\mathcal{X})$. Assume that the infimum eq:variational_problem is fin

Figures (6)

  • Figure 1: An illustration of two different constructions, showing which two-dimensional marginals of $\Upsilon_n$ are optimal. Here, $n=2$ and the time interval $[0,1]$ is divided into $2^2=4$ equal pieces.
  • Figure 2: Two cases in the computation of $b^{\alpha,p}$-regularity of a piecewise geodesic curve $X^n$ in the proof of \ref{['lemma:balphap_Xn']}. Note that $n$ is fixed. Top:$D_m$ is a coarser partition than $D_n$. Bottom:$D_m$ is a finer partition than $D_n$.
  • Figure 3: \ref{['exp:infinitesimal_variation']} yields a compatible $(\mu_t) \in C^{\upgamma \textrm{-} \mathrm{H\ddot{o}l}} ([0,1];P_p(\mathbb{R}^2))$ from superposition of absolutely continuous curves. Top: The initial measure $\mu_0$ consists of a countable family of particles indexed by $j$, whose mass $w_j$ decreases with respect to $j$. As time runs, the particles oscillate vertically between 0 and 1. The lighter a particle is, the faster it moves. Their $x$-coordinates, which are separated by a distance $a>1$, remain constant. Bottom:$y$-coordinates of the first three particles as a function of time, as defined in \ref{['eq:counterexample_yj']}. Each curve has constant speed for a.e. $t \in [0,1]$.
  • Figure 4: \ref{['exp:compatibility_vs_noncompatibility']} yields non-compatible piece-wise geodesic curves $(\mu_t^j) \in C ([0,1];P_p(\mathbb{S}^1))$ indexed by $j$ by modifying a constant-speed geodesic $(\mu_t)$ in the $p$-Wasserstein space on $\mathbb{S}^1$, where $p\in [1,\infty)$. Top: Initial measures defined in \ref{['eq:compatibility_mu']} and \ref{['eq:noncompatibility_mu_j']}. Each $\mu_t^j$ consists of $2^{j+1}$ particles with equal mass arranged equidistantly on the circle with a perimeter equal to 2. As time runs, the particles rotate around the circle with constant speed 1. Bottom: Wasserstein distance from the initial measure as a function of time. The metric speed of the Wasserstein curves is 1 for a.e. $t \in [0,1]$ for all $j$, whereas the fractional Sobolev norm tends to $0$ as $j \to \infty$.
  • Figure 5: \ref{['exp:compatibility_vs_noncompatibility']}. The measure $(\mu^{j=0}_t)$ defined in \ref{['eq:noncompatibility_mu_j']} at 4 time points. The collections at times $\{0,\frac{1}{4},\frac{2}{4}\}$ and $\{\frac{1}{4}, \frac{2}{4}, \frac{3}{4}\}$ are compatible, but not at $\{0,\frac{1}{4},\frac{2}{4},\frac{3}{4}\}$. To confirm the compatibility, checking the optimality only for three measures is not enough.
  • ...and 1 more figures

Theorems & Definitions (98)

  • Proposition 1.1: Existence of a minimizer
  • Theorem 1.2
  • Theorem 1.3: Existence of a minimizing lift
  • Proposition 1.4
  • Definition 1.5: Compatibility of measures in $P_p(\mathcal{X})$
  • Theorem 1.6: Construction of a realizing lift
  • Remark 1.7: A weaker compatibility condition
  • Corollary 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 88 more