On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes

Alexander Nazarov,Yakov Nikitin

doi:10.3390/math6040055

Abstract

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.

Highlights

The problem of small deviation asymptotics for Gaussian processes was intensively studied in last years
We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes
We provide the exact small deviation asymptotics for more complicated mixtures containing the Ornstein–Uhlenbeck processes

Summary

Introduction

The problem of small deviation asymptotics for Gaussian processes was intensively studied in last years. The small deviations of the process Y(Hβ) were studied at the logarithmical level in [16], where the following result was obtained We cite it in the simplified form (without the weight function). Let X1 and X2 be independent zero mean Gaussian processes on [0, 1] We assume that their covariance functions G1 (s, t) and G2 (s, t) are the Green functions for the same ODO (1) with different boundary conditions. It is the Green function for the ODO 1+1β2 L subject to some (in general, more complicated) boundary conditions This allows us to apply general results of [6,8] on the small ball behavior of the Green Gaussian processes and to obtain the asymptotics of P{|| Z β ||2 ≤ ε}. The following asymptotic relation holds as ε → 0: P{||B(α) + βU(α) ||2 ≤ ε} ∼

Mixed Processes Related to Polynomials of Covariance Operator

Discussion