Fractional Brownian Fields over Manifolds

TL;DR

This paper extends fractional Brownian fields to complete Riemannian manifolds for all Hurst parameters, establishing their existence, properties, and stationary versions, with implications for stochastic analysis on manifolds.

Contribution

It introduces a novel construction of fractional Brownian fields over manifolds valid for all Hurst parameters, including their properties and stationary variants.

Findings

Existence of fractional Brownian fields on manifolds for all b5 b7 (0,1)

Proved distributional self-similarity and stationarity of increments

Established almost sure Hb6lder continuity of sample paths

Abstract

Extensions of the fractional Brownian fields are constructed over a complete Riemannian manifold. This construction is carried out for the full range of the Hurst parameter $\alpha\in(0,1)$. In particular, we establish existence, distributional scaling (self-similiarity), stationarity of the increments, and almost sure H\"{o}lder continuity of sample paths. Stationary counterparts to these fields are also constructed.