Higher-dimensional local existence of fractional Lévy fields

Establish the local existence of fractional Lévy fields with structure function d(η,ζ)^{2H} on a broad class of higher-dimensional Riemannian manifolds, extending the known local existence result for two-dimensional manifolds and the constructions available in special model spaces.

Background

The paper constructs fractional Lévy fields locally on arbitrary two-dimensional Riemannian manifolds by using a measure on geodesics and then applying a Schoenberg-type argument for Hurst parameters below one-half. The authors explain that the geodesic-measure construction does not naturally extend to higher dimensions except in special settings such as model spaces.

Consequently, the local existence problem for fractional Lévy fields on general higher-dimensional Riemannian manifolds remains unresolved. The problem is explicitly framed as an open problem at the end of the appendix.