Exact asymptotics in eigenproblems for fractional Brownian covariance operators

TL;DR

This paper develops a spectral analysis framework for fractional covariance operators, providing precise asymptotics for eigenvalues and eigenfunctions, which advances understanding of fractional Gaussian processes and related probabilistic and physical problems.

Contribution

It introduces a novel approach for asymptotic eigenanalysis of fractional covariance operators, enabling solutions to longstanding problems in fractional Gaussian process theory.

Findings

Derived accurate asymptotic formulas for eigenvalues and eigenfunctions.

Enabled computation of exact small ball probability limits for fractional processes.

Analyzed asymptotics of integral equations in physics and probability.

Abstract

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.