2023/05/19 by Alberto González-Sanz, Marc Hallin, González-Sanz, Alberto +3 · 1 citation
Mathematics · #49Q22 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Nonlinear Partial Differential Equations #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2305.11751
openalex publication_date 2023/05/19 · openalex created_date 2023/05/24 · openalex updated_date 2026/07/28
The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures \rm P and \rm Q on a separable Hilbert space H where \rm P does not give mass to "small sets" (namely, Lipschitz hypersurfaces), we show, without imposing any moment assumptions, that there exists a gradient of convex function ∇ψ pushing \rm P forward to \rm Q. In case H is infinite-dimensional, \rm P-a.s. uniqueness is not guaranteed, though. If, however, \rm Q is boundedly supported (a natural assumption in several statistical applications), then this gradient is \rm P a.s. unique. In the second part of the paper, we establish stability results for transport maps in the sense of uniform convergence over compact "regularity sets". As a consequence, we obtain a central limit theorem for the fluctuations of the optimal quadratic transport cost in a separable Hilbert space.