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

Let (W,μ,H)(W,μ,H) be an abstract Wiener space and assume that YY is a signal of the form Y=X+wY=X+w, where XX is an HH-valued random variable, ww is the generic element of WW. Under the hypothesis of independence of ww and XX, we show that the quadratic estimate of XX, denoted by Xˆ(Y)=E[X|Y], is of the form ∇F(Y)∇F(Y), where FF is an HH-convex function on WW. We prove also some relations between the quadratic estimate error and the Wasserstein distance between some natural probabilities induced by the shift IH+∇FIH+∇F and the conditional law of YY given XX.