On the existence of fractional Brownian fields indexed by manifolds with closed geodesics

TL;DR

This paper establishes geometric conditions for the existence of fractional Brownian fields on manifolds, revealing that certain topological properties like simple connectivity are necessary for the Le9vy Brownian field to exist.

Contribution

It provides necessary geometric conditions for fractional Brownian fields on manifolds, highlighting the topological restrictions for the Le9vy Brownian field, especially on non-simply connected manifolds.

Findings

Compact manifolds with minimal closed geodesics are simply connected.

Le9vy Brownian fields do not exist on hyperbolic spaces.

Alternative kernels are needed for Gaussian modeling on non-simply connected manifolds.

Abstract

We give a necessary condition of geometric nature for the existence of the $H$-fractional Brownian field indexed by a Riemannian manifold. In the case of the L\'evy Brownian field ($H=1/2$) indexed by manifolds with minimal closed geodesics it turns out to be very strong. In particular we show that compact manifolds admitting a L\'evy Brownian field are simply connected. We also derive from our result the nondegenerescence of the L\'evy Brownian field indexed by hyperbolic spaces. These results stress the need for alternative kernels on nonsimply connected manifolds to allow for Gaussian modelling or kernel machine learning of functional data with manifold-valued entries.