Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

TL;DR

This paper derives two-term eigenvalue asymptotics for integro-differential equations related to fractional Gaussian processes, advancing understanding of their spectral properties and applications to small ball probabilities.

Contribution

It generalizes the Chigansky--Kleptsyna method to a broader class of equations, providing new spectral asymptotics relevant to Gaussian process theory.

Findings

Established two-term eigenvalue asymptotics for the equations

Applied results to small ball probability estimates in L2-norm

Extended spectral analysis techniques to fractional Gaussian processes

Abstract

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky--Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in $L_2$-norm is given.