On the eigenproblem for Gaussian bridges

TL;DR

This paper investigates the spectral properties of Gaussian bridges derived from base processes, providing asymptotic results for their eigenvalues, including for fractional Brownian motion.

Contribution

It offers new insights into the eigenproblem for Gaussian bridges, linking the spectrum of the bridge to that of the base process asymptotically.

Findings

Asymptotic spectral characterization for Gaussian bridges

Results applicable to fractional Brownian motion

Enhanced understanding of covariance operator spectra

Abstract

Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately it is notoriously hard to derive in a closed form. In this paper we consider the eigenproblem for Gaussian bridges. Given a {\em base} process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.