Averaging of random walks and shift-invariant measures on a Hilbert space

We study random walks in a Hilbert space $H$ and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on $H$. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure $\lambda$ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space $H$ containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in $H$. We define the Hilbert space $\mathcal H$ of equivalence classes of complex-valued functions on $H$ that are square integrable with respect to a shift-invariant measure $\lambda$. Using averaging of the shift operator in $\mathcal H$ over random vectors in $H$ with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on $H$, we define a one-parameter semigroup of contracting self-adjoint transformations on $\mathcal H$, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.