Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations

We study random walks in a Hilbert space $H$ and their applications to representations of solutions to Cauchy problems for differential equations whose initial conditions are numerical functions on the Hilbert space $H$. Examples of such representations of solutions to various evolution equations in the case of a finite-dimensional space $H$ are given. Measures on a Hilbert space that are invariant with respect to shifts are considered for constructing such representations in infinite-dimensional Hilbert spaces. According to a theorem of A. Weil, there is no Lebesgue measure on an infinite-dimensional Hilbert space. We study a finitely additive analog of the Lebesgue measure, namely, a nonnegative, finitely additive measure $\lambda$ defined on the minimal ring of subsets of an infinite-dimensional Hilbert space $H$ containing all infinite-dimensional rectangles whose products of sides converge absolutely; this measure is invariant with respect to shifts and rotations in the Hilbert space $H$. We also consider finitely additive analogs of the Lebesgue measure on the spaces $l_{p}$, $1\leq p\leq \infty$, and introduce the Hilbert space $\mathcal H$ of complex-valued functions on the Hilbert space $H$ that are square integrable with respect to a shift-invariant measure $\lambda$. We also obtain representations of solutions to the Cauchy problem for the diffusion equation in the space $H$ and the Schrödinger equation with the coordinate space $H$ by means of iterations of the mathematical expectations of random shift operators in the Hilbert space $\mathcal H$.