Annealed quantitative estimates for the quadratic 2D-discrete random matching problem

• 成果类型: Article
• 署名作者: Clozeau, Nicolas; Mattesini, Francesco
• 署名单位: Institute of Science & Technology - Austria; University of Munster; Max Planck Society
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01254-0
• 发表日期: 2024
• 页码: 485-541
• 关键词: wasserstein distance; polar factorization; CONVERGENCE

摘要

We study a random matching problem on closed compact 2-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m = m(n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures mu(n) and v(m) is quantitatively well-approximated by (Id, exp(del h(n)))(#) mu(n) where h(n) solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Ampere equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the alpha-mixing coefficient holds and for a class of discrete-time sub-geometrically ergodic Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure.