Linear Multifractional Stable Sheets in the Broad Sense: Existence and Joint Continuity of Local Times

TL;DR

This paper introduces linear multifractional stable sheets (LMSS) encompassing both Brownian and stable sheets, and establishes conditions for their local times' existence, continuity, and H"older regularity, advancing understanding of their sample path properties.

Contribution

It provides the first comprehensive conditions for the existence and joint continuity of local times of LMSS, extending previous results to a broader class of processes.

Findings

Necessary and sufficient condition for local times existence

Sufficient condition for joint continuity of local times

Sharp local H"older condition in the set variable

Abstract

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $\alpha\in(0,2]$, to include both linear multifractional Brownian sheets ($\alpha=2$) and linear multifractional stable sheets ($\alpha<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local H\"older condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint continuity of local times; and Theorem 4.1 proves a sharp local H\"older condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.