Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups

We study measures on a real separable Hilbert space $E$ that are invariant under translations by arbitrary vectors in $E$. We define the Hilbert space $\mathcal H$ of complex-valued functions on $E$ square-integrable with respect to some translation-invariant measure $\lambda$. We determine the expectations of the operators of shift by random vectors whose distributions are given by semigroups (with respect to convolution) of Gaussian measures on $E$. We prove that these expectations form a semigroup of self-adjoint contractions on $\mathcal H$. We obtain a criterion for the strong continuity of such semigroups and study the properties of their generators (which are self-adjoint generalizations of Laplace operators to the case of functions of infinite-dimensional arguments). We introduce analogues of Sobolev spaces and spaces of smooth functions and obtain conditions for the embedding and dense embedding of spaces of smooth functions in Sobolev spaces. We apply these function spaces to problems of approximating semigroups by the expectations of random processes and study properties of our generalizations of Laplace operators and their fractional powers.