Local times of linear multifractional stable sheets

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

“…In Theorem 2.11, we provide a sufficient condition for the joint continuity of the local times of LMSS, which is significantly weaker than the conditions proved in [14] for multifractional Brownian sheets and in [19] for linear multifractional stable sheets. We remark that [19] makes crucial use of the arguments in [25], which relies on the assumption of α ∈ (1, 2). Our Theorem 2.11 holds for all α ∈ (0, 2].…”

“…We will use the notion of linear multifractional stable sheets in the broad sense (LMSS), where the stability index is α, which controls the tail-heaviness of the distributions, is ranged in (0, 2]. As a consequence of the present paper, some results obtained in [1,4,5,7,14,19,22,25] are extended to the setting of LMSS and improved significantly. Below we describe the main contributions of this paper: (i).…”

“…The purpose of this paper is to develop a unified framework for improving the results concerning the local times of multifractional Brownian sheets [14], linear fractional stable sheets [1], and linear multifractional stable sheets [19]. We will use the notion of linear multifractional stable sheets in the broad sense (LMSS), where the stability index is α, which controls the tail-heaviness of the distributions, is ranged in (0, 2].…”

“…Below we describe the main contributions of this paper: (i). A sufficient condition for the existence of local times of multifractional Brownian sheets was given in [14], which was extended to linear multifractional stable sheets in [19]. However, their conditions are not optimal.…”

“…In particular, no necessary condition had been proved for the existence of local times of linear multifractional Brownian or linear multifractional stable sheets in the literature. We fill this gap by proving in Theorem 2.9 a sufficient and necessary condition for the existence of the local times of LMSS, which solves completely the problem on the existence of local times for multifractional Brownian sheets in [14] and linear multifractional stable sheets in [19]. (ii).…”