Nonexistence of fractional Brownian fields indexed by cylinders

TL;DR

This paper proves the nonexistence of fractional Brownian fields on cylinders and similar spaces, revealing limitations of classical kernels for Gaussian modeling on such metric spaces.

Contribution

It demonstrates that fractional Brownian fields cannot be defined on cylinders and extends this to Riemannian products, highlighting challenges in kernel construction on these spaces.

Findings

No fractional Brownian field exists on cylinders for any H>0.

The kernel d^{2H} is not negative definite on cylinders, affecting Gaussian process modeling.

Discontinuity of negative definiteness set under Gromov-Hausdorff convergence.

Abstract

We show in this paper that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S}^1 \times ]0,\varepsilon[$ endowed with its product distance $d$ for any $\varepsilon>0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels -- or covariances -- on the cylinder. We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle. Another consequence is the discontinuity of the set of $H$ such that $d^{2H}$ is negative definite with respect to the Gromov-Hausdorff convergence on compact metric spaces. These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.