Nonexistence of fractional Brownian fields indexed by cylinders

Nil Venet

Published 2016-12-18Version 1

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 for any $\varepsilon>0$ and $H>0$. As a consequence we show that the fractional index of a compact metric space is not continuous with respect to the Gromov-Hausdorff convergence. The result holds for Riemannian products 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.