Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability

Abstract: 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 $\mathcal{H}$ 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 $\nabla\psi$ pushing ${\rm P} $ forward to ${\rm Q}$. In case $\mathcal{H}$ 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.