L2-small ball asymptotics for Gaussian random functions: A survey

Alexander Nazarov; Yulia Petrova

doi:10.1214/23-PS20

2023

L_{2}

-small ball asymptotics for Gaussian random functions: A survey

Alexander Nazarov, Yulia Petrova

Author Affiliations +

Probab. Surveys 20: 608-663 (2023). DOI: 10.1214/23-PS20

Abstract

This article is a survey of the results on asymptotic behavior of small ball probabilities in $L_{2}$ -norm. Recent progress in this field is mainly based on the methods of spectral theory of differential and integral operators.

Funding Statement

The work was supported by Russian Science Foundation, Grant 21-11-00047. The most part of this paper was written while Yu.P. was a postdoctoral fellow at IMPA. She thanks IMPA for creating excellent working conditions.

Dedication

Dedicated to the memory of M.S. Birman and M.Z. Solomyak,founders of St. Petersburg school of spectral theory

Acknowledgments

We are deeply grateful to M.A. Lifshits who suggested us the idea of this survey, and to I.A. Sheipak who provided us with some important references.

Citation

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Alexander Nazarov. Yulia Petrova. " $L_{2}$ -small ball asymptotics for Gaussian random functions: A survey." Probab. Surveys 20 608 - 663, 2023. https://doi.org/10.1214/23-PS20