Analogs of the Lebesgue Measure and Diffusion in a Hilbert Space

Abstract

We study shift-invariant measures on a real separable Hilbert space E, which are also invariant with respect to orthogonal transforms. In this article a finitely additive analogue of the Lebesgue measure is constructed. It is a non-negative finitely additive shift-invariant measure on a special ring of subsets from a space E, which is invariant with respect to orthogonal transforms. The ring contains all infinite-dimensional rectangles, whose products of side lengths are absolutely convergent. We define a Hilbert space \( \mathcal{\mathscr{H}} \), whose elements are complex-valued functions. The functions are square-integrable by some shift-invariant measure, which is also invariant with respect to rotations. We define the expected values of shift operators over random vectors, whose distribution is given by a family of Gaussian measures on a space E. The measures form a semigroup with respect to convolution. We prove that the expected values form a semigroup of contracting self-adjoint operators on a space \( \mathcal{\mathscr{H}} \). The semigroup is not strongly continuous. We also find invariant subspaces, where the semigroup is continuous in the strong operator topology. We investigate the structure of arbitrary (not necessarily continuous) operator semigroups of contracting self-adjoint transforms on a Hilbert space. We show that the suggested in Orlov et al. (Izv. Math. 80(6), 1131–1158 2016) method of the Feynman averaging is applicable to discontinuous semigroups.