Linear multifractional stable sheets in the broad sense: Existence and joint continuity of local times

Abstract

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $\mathit{\alpha }\in (0,2]$, to include both linear multifractional Brownian sheets ($\mathit{\alpha }=2$) and linear multifractional stable sheets ($\mathit{\alpha }<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder 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 Hölder 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.