On exact Hausdorff measure functions of operator semistable Lévy processes

ABSTRACT

Let X = {X(t)}_{t ⩾ 0} be an operator semistable Lévy process on with exponent E, where E is an invertible linear operator on . In this article, 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.

KEYWORDS:

• Operator semistable Lévy process
• sample path
• range
• semi-selfsimilarity
• exact Hausdorff measure
• gauge function
• expected sojourn time
• semistable subordinator
• Tauberian theorem

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G51
• 28AG78
• 28A80
• 60G17
• 60G18
• 60G52

Funding

This work was supported by Deutsche Forschungsgemeinschaft (DFG) under grant KE1741/6-1.