Joint continuity of the local times of linear fractional stable sheets

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 = (H₁, ..., 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 < H₁^-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 H₁, ..., 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).