Abstract. Let X = {X(t)} t≥0 be an operator semistable Lévy process in R d with exponent E, where E is an invertible linear operator on R d and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao [17] for the special case of an operator stable (selfsimilar) Lévy process, for an arbitrary Borel set B ⊆ R + we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.

Abstract. Let X = {X(t) : t ≥ 0} be an operator semistable Lévy process in R d with exponent E, where E is an invertible linear operator on R d . For an arbitrary Borel set B ⊆ R + we interpret the graph Gr X (B) = {(t, X(t)) : t ∈ B} as a semi-selfsimilar process on R d+1 , whose distribution is not full, and calculate the Hausdorff dimension of Gr X (B) in terms of the real parts of the eigenvalues of the exponent E and the Hausdorff dimension of B. We use similar methods as applied in [12] and [6].

Abstract. Let X = {X(t)} t≥0 be an operator semistable Lévy process on R d with exponent E, where E is an invertible linear operator on R d . In this paper we determine exact Hausdorff measure functions for the range of X over the time interval [0, 1] under certain assumptions on the principal spectral component of E.As a byproduct we also present Tauberian results for semistable subordinators and sharp bounds for the asymptotic behavior of the expected sojourn times of X.

Abstract. Dilative semistability extends the notion of semi-selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. It is shown that this scaling relation is a natural extension of dilative stability and some examples of dilatively semistable processes are given. We further characterize dilatively stable and dilatively semistable processes as limits for certain rescaled aggregations of independent processes.

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