Exact L_2-Small Deviation Asymptotics for Some Brownian Functionals

TL;DR

This paper derives exact small deviation asymptotics for certain Gaussian processes with respect to weighted Hilbert norms, introducing a method that bypasses the need for eigenfunction knowledge and generalizes previous results.

Contribution

It presents a novel approach to obtain exact small deviation asymptotics without requiring eigenfunctions, extending results to Brownian excursions, meanders, and Bessel processes.

Findings

Exact asymptotics for Gaussian processes with weighted norms

New relations for Brownian excursion and meander deviations

Generalization of previous small deviation results

Abstract

We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.