Estimation for the additive Gaussian channel and Monge-Kantorovitch measure transportation

Let (W, mu, H) be an abstract Wiener space and assume that Y is a signal of the form Y = X + w, where X is an H-valued random variable, w is the generic element of W. Under the hypothesis of independence of w and X, we show that the quadratic estimate of X, denoted by (X) over cap (Y) = E [X|Y], is of the form del F (Y), where F is an H-convex function on W. We prove also some relations between the quadratic estimate error and the Wasserstein distance between some natural probabilities induced by the shift I-H + del F and the conditional law of Y given X. (C) 2007 Elsevier B.V. All rights reserved.