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

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 P and Q on a separable Hilbert space H where P doesnot 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 P forward to Q. In case H is infinite-dimensional, P-a.s. uniqueness is not guaranteed, though. If, however, Q is boundedly supported (a natural assumption in several statistical applications), then this gradient is P-a.s. unique. In the second part of the paper, we establish stability results for the same 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.