Derivatives of local times for some Gaussian fields

TL;DR

This paper investigates the existence and regularity of derivatives of local times for a class of Gaussian fields, establishing conditions for their existence and continuity, with implications for Gaussian process analysis.

Contribution

It provides new conditions for the existence of derivatives of local times in Gaussian fields and proves their Hölder continuity under these conditions.

Findings

Derived conditions for local time derivatives existence

Proved Hölder continuity of derivatives in time and space

Established necessity of conditions at the origin

Abstract

In this article, we consider derivatives of local time for a $(2,d)$-Gaussian field \[ Z=\big\{ Z(t,s)= X^{H_1}_t -\widetilde{X}^{H_2}_s, s,t \ge 0\big\}, \] where $X^{H_1}$ and $\widetilde{X}^{H_2}$ are two independent processes from a class of $d$-dimensional centered Gaussian processes satisfying certain local nondeterminism property. We first give a condition for existence of derivatives of the local time. Then, under this condition, we show that derivatives of the local time are H\"{o}lder continuous in both time and space variables. Moreover, under some additional assumptions, we show that this condition is also necessary for existence of derivatives of the local time at the origin.