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 , is of the form [backward difference]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 IH+[backward difference]F and the conditional law of Y given X.