Krein’s spectral theory and the Paley–Wiener expansion for fractional Brownian motion

Abstract

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein’s work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley–Wiener expansion of the ordinary Brownian motion to the fractional case.