Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure

We study measures on a real separable Hilbert space $E$ that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of $E$ that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $\mathcal H$ of complex-valued functions on $E$ that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on $E$ that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $\mathcal H$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.