Regularity of the conditional expectations with respect to signal to noise ratio

Let $(W,H,\mu )$ be the classical Wiener space, assume that $U_{\text{la}}=I_W+u_{\text{la}}$ is an adapted perturbation of identity where the perturbation $u_{\text{la}}$ is an $H$-valued map, defined up to $\mu$-equivalence classes, such that its Lebesgue density $s\to {\dot{u}}_{\text{la}}(s)$ is almost surely adapted to the canonical filtration of the Wiener space and depending measurably on a real parameter $\text{la}$. Assuming some regularity for $u_{\text{la}}$, its Sobolev derivative and integrability of the divergence of the resolvent operator of its Sobolev derivative, we prove the almost sure and $L^p$-regularity w.r. to $\text{la}$ of the estimation $E[{\dot{u}}_{\text{la}}(s)\mathrm{\mid }{\text{calU}}_{\text{la}}(s)]$ and more generally of the conditional expectations of the type $E[F\mid {\text{calU}}_{\text{la}}(s)]$ for nice Wiener functionals, where $({\text{calU}}_{\text{la}}(s),s\in [0,1])$ is the the filtration which is generated by $U_{\text{la}}$. These results are applied to prove the invertibility of the adapted perturbations of identity, hence to prove the strong existence and uniqueness of functional SDE’s; convexity of the entropy and the quadratic estimation error and finally to the information theory.