Exact L_2-small ball asymptotics of Gaussian processes and the spectrum of boundary value problems with "non-separated" boundary conditions

TL;DR

This paper refines classical spectral asymptotics for differential operators to derive exact small ball probabilities for a new class of Gaussian processes, including integrated Wiener and Brownian bridge processes.

Contribution

It provides the exact $L_2$-small ball asymptotics for a broad class of Gaussian processes using sharpened spectral asymptotics of boundary value problems.

Findings

Exact $L_2$-small ball asymptotics for new Gaussian process classes

Includes integrated generalized Slepian process and Brownian bridge

Refines classical spectral asymptotic results

Abstract

We sharpen a classical result on the spectral asymptotics of the boundary value problems for self-adjoint ordinary differential operator. Using this result we obtain the exact $L_2$-small ball asymptotics for a new class of zero mean Gaussian processes. This class includes, in particular, integrated generalized Slepian process, integrated centered Wiener process and integrated centered Brownian bridge.