On the Variational Regularity of Cameron-Martin paths

TL;DR

This paper establishes a general result linking the regularity of Gaussian process covariance to the Cameron-Martin paths, extending previous results especially for fractional Brownian motion, with implications for rough path theory.

Contribution

It provides a unified theorem that improves understanding of the regularity of Cameron-Martin paths for a broad class of Gaussian processes, including fractional Brownian motion.

Findings

Finite rho-variation of covariance implies finite rho-variation of Cameron-Martin paths

Sharpened regularity results for fractional Brownian motion in rougher regimes

Highlights the significance of these regularity properties for applications in stochastic analysis

Abstract

It is a well-known fact that finite rho-variation of the covariance (in 2D sense) of a general Gaussian process implies finite rho-variation of Cameron-Martin paths. In the special case of fractional Brownian motion (think: 2H=1/rho), in the rougher than Brownian regime, a sharper result holds thanks to a Besov-type embedding [Friz-Victoir, JFA, 2006]. In the present note we give a general result which closes this gap. We comment on the importance of this result for various applications.