Sample Path Properties of a Class of Operator Stable Processes

On exact Hausdorff measure functions of operator semistable Lévy processes

“…Let α 1 and d 1 be as defined in Section 2.2 by means of the spectral decomposition. As in [8] we were only able to fully solve the question of exact Hausdorff measures for the range of operator semistable Lévy processes in the case α 1 < d 1 but also give partial results for the case α 1 > d 1 . We will consider operator semistable Lévy processes of type A and type B, simultaneously.…”

“…For an arbitrary Borel measure µ on R d and a function φ ∈ Φ, the upper φ-density of µ at x ∈ R d is defined as where B(x, r) denotes the closed ball with radius r centered at x. The following lemma is similar to Lemma 2.1 in [8] and is a direct consequence of the results in [18].…”

“…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.properties of Lévy processes with independent stable components, including exact Hausdorff measures. Based on their work, Hou and Ying [8] determined an exact Hausdorff measure function for the range of certain operator stable Lévy process of type A with diagonal exponent E. They emphasize without proof that similar methods also lead to an exact Hausdorff measure function for type B, see Remark 1 in [8]. Corresponding results for the graph are presented in [7].…”

“…Firstly, we consider the more general class of operator semistable Lévy process with the weaker discrete scaling (1.1), secondly, we will relax the assumption that the exponent E should be diagonal and show that it suffices to require diagonality for a principal spectral component; see Section 2.2 for details. Lastly, we will also derive exact Hausdorff measure functions for type B which turned out to be more challenging than asserted in Remark 1 of [8]. The Hausdorff dimension for the range and the graph of operator semistable Lévy processes have recently been determined in [11] and [23], respectively; see also [10] for an alternative derivation based on an index formula presented in [14].…”

“…properties of Lévy processes with independent stable components, including exact Hausdorff measures. Based on their work, Hou and Ying [8] determined an exact Hausdorff measure function for the range of certain operator stable Lévy process of type A with diagonal exponent E. They emphasize without proof that similar methods also lead to an exact Hausdorff measure function for type B, see Remark 1 in [8]. Corresponding results for the graph are presented in [7].…”