Derivatives of local times for some Gaussian fields II

TL;DR

This paper investigates the conditions under which derivatives of the local time exist for a specific class of Gaussian fields formed by differences of independent Gaussian processes with certain properties.

Contribution

It provides necessary conditions for the existence of derivatives of local times for a class of $(2,d)$-Gaussian fields constructed from independent Gaussian processes.

Findings

Identifies necessary conditions for local time derivatives existence

Focuses on Gaussian fields formed by differences of independent processes

Advances understanding of local time regularity for Gaussian fields

Abstract

Given a $(2,d)$-Gaussian field \[ Z=\big\{ Z(t,s)= X^{H_1}_t -\tilde{X}^{H_2}_s, s,t \ge 0\big\}, \] where $X^{H_1}$ and $\tilde{X}^{H_2}$ are independent $d$-dimensional centered Gaussian processes satisfying certain properties, we will give the necessary condition for existence of derivatives of the local time of $Z$.