Moments estimates for local times of one class of Gaussian processes

TL;DR

This paper establishes the existence and provides moments estimates for the local times of Gaussian integrators, linking their properties to associated linear operators and improving previous results.

Contribution

It introduces conditions on non-invertible operators ensuring local time existence and derives moments estimates, enhancing understanding of Gaussian integrators' local times.

Findings

Local time exists under specific operator conditions.

Moments estimates for local times are derived.

Local time depends continuously on the generating operator.

Abstract

In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [1].