Joint Continuity Of Local Times Research Articles

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

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with α∈(0,2], to include both linear multifractional Brownian sheets (α=2) and linear multifractional stable sheets (α<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.

A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times.

Some results on fractional Brownian sheets and their local times

Let B 0 = {B 0 (t), t ∈ R + } be a real-valued fractional Brownian sheet. Define the (N, d)-Gaussian random field B H by B H (t) = (B 1 (t), ..., B (t)) t ∈ R + , where B 1 , ..., B are independent copies of B 0 . The existence and joint continuity of local times of B H is proven in some given conditions in [22]. We then study further properties of the local times of B H , such as the moments of increments of local times, the large increments and the maximum moduli of continuity of local times and as a result, we answer the questions posed in [22].