Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2502.12068

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

Ehsan Abedi
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.MGmath.OCmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ehsan Abedi. 2025-10-01. Fractional Sobolev paths on Wasserstein spaces and their energy-minimizing particle representations. https://arxiv.org/abs/2502.12068

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗
arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗