Diffusion in Hilbert space endowing with traslationary rotationary invariant measure

A measures on real separable Hilbert space which is invariant with respect to the shift on any vector and to any orthogonal mapping are studied. The mean values of operator of a shift on random vector with Gaussian distributions are investigated. It had been proved that the family of mean values of random shift operators is the semigroup of self-adjoint contractions which is not strongly continuous. The structure of general semigroup of self-adjoint contractions in Hilbert space without the strong continuity property had been described.