Joint continuity of the local times of linear fractional stable sheets

Abstract

Linear fractional stable sheets (LFSS) are a class of random fields containing the class of fractional Brownian sheets (FBS) by allowing, in the linear fractional representation of the FBS, the random measure to be α-stable with α ∈ ( 0 , 2 ] . In this Note, we extend some properties of the local time shown in the Gaussian case to the symmetric α-stable case. For any N ⩾ 1 , an ( N , 1 ) -LFSS is a real valued random field defined on R + N . When N = 1 , the process is called linear fractional stable motion (LFSM). For N ⩾ 1 , an ( N , 1 ) -LFSS is mainly parameterized by a multidimensional index H = ( H 1 , … , H N ) ∈ ( 0 , 1 ) N . Let N , d ⩾ 1 be fixed, we consider a random field defined on R + N and taking its values in R d , an ( N , d ) -LFSS, whose components are d independent copies of the same ( N , 1 ) -LFSS. We show that, if d < H 1 −1 + ⋯ + H N −1 , then the ( N , d ) -LFSS with index H has a local time. Moreover, when the sample path of the LFSS is continuous, that is, for α < 2 , when H 1 , … , H N > 1 / α , we show that the local time is jointly continuous. To cite this article: A. Ayache et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).