On probabilistic commutative spaces (Q1119915)

In 1960, \textit{P. H. Roberts} and \textit{H. D. Ursell} [Philos. Trans. R. Soc. Lond., A 252, 317-356 (1960; Zbl 0094.319)] worked out an interesting theory of ``commutative'' Riemannian spaces as those compact Riemannian manifolds on which ``every two random steps commute''. In particular, they have proved: 1) A compact analytic Riemannian manifold is commutative if and only if all Euclidean Laplacians on it commute. 2) Each (compact) commutative space has volume-preserving local geodesic symmetries. In the present paper, the earlier results are generalized to the non- compact case, and a completely new characterization of these spaces is given which uses an infinite set of natural two-point functions.