Stochastic regularization of linear equations and the realization of Gaussian fields

AbstractThe variational representation of the conditional expectation X̂ of a Gaussian signal X given observations Y corrupted by independent white noise is investigated in the general infinite-dimensional setting. Under Hilbert-Schmidt type assumptions it is shown that the filter X̂ can be realized on sample configurations Yω as the extension by continuity of the mapping that gives the solution of a related variational problem