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

Let ( W , μ , 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 ˆ ( Y ) = E [ X | Y ] , is of the form ∇ 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 + ∇ F and the conditional law of Y given X .