Random Walks and Measures on Hilbert Space that are Invariant with Respect to Shifts and Rotations

Abstract

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 number-valued 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 λ 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 ≤ p ≤ ∞, 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 λ. 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 $\mathcal{H}$.