Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

TL;DR

This paper derives the small deviation asymptotics for Gaussian processes with respect to degenerate self-similar measures, extending understanding of their probabilistic behavior in specialized measure spaces.

Contribution

It introduces new asymptotic results for Gaussian processes under degenerate self-similar measures, including classical processes like Brownian motion.

Findings

Logarithmic small ball asymptotics established for Gaussian processes

Includes processes such as Brownian motion and Ornstein-Uhlenbeck

Provides insights into spectral properties and deviations

Abstract

We find the logarithmic small ball asymptotics for the $L_2$-norm with respect to a degenerate self-similar measures of a certain class of Gaussian processes including Brownian motion, Ornstein - Uhlenbeck process and their integrated counterparts.