Small Ball Probabilities for Smooth Gaussian fields and Tensor Products of Compact Operators

TL;DR

This paper derives small ball probability asymptotics for smooth Gaussian fields with tensor product covariance structures, extending spectral analysis techniques for compact operators.

Contribution

It introduces new spectral asymptotics for tensor products of compact self-adjoint operators, broadening understanding of Gaussian fields with tensor product covariances.

Findings

Established logarithmic $L_2$-small ball asymptotics for a class of Gaussian fields.

Extended spectral asymptotics to include smooth covariances and classical stochastic processes.

Developed a new theorem on spectral asymptotics for tensor products of compact operators.

Abstract

We find the logarithmic $L_2$-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein--Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper \cite{KNN}, where the regular behavior at infinity of marginal eigenvalues was assumed.