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 undertranslations by arbitrary vectors in $E$. We define the Hilbert space $\mathcal H$ ofcomplex-valued functions on $E$ square-integrable with respect to some translation-invariant measure $\lambda$. We determine the expectations of the operators of shiftby random vectors whose distributions are given by semigroups (with respect toconvolution) 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 thestrong continuity of such semigroups and study the properties of their generators(which are self-adjoint generalizations of Laplace operators to the case of functionsof infinite-dimensional arguments). We introduce analogues of Sobolev spaces andspaces of smooth functions and obtain conditions for the embedding and dense embedding ofspaces of smooth functions in Sobolev spaces. We apply these function spacesto problems of approximating semigroups by the expectations of random processes andstudy properties of our generalizations of Laplace operators and their fractionalpowers.

translation-invariant measure on a Hilbert space, Laplace operator on an infinite-dimensional space, Sobolev spaces, embedding theorems, random walks