Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties (Q1745280)

Let \({X=\{X(t)\}_{t\geq0}}\) be a Lévy process in \(\mathbb{R}^d\) and \(\psi:\mathbb{R}^d\to\mathbb{C}\) its Lévy exponent, i.e. it is a continuous function with \({\psi(0)=0}\) defined by \[ {\mathbb{E}(e^{i(\xi,X(t))})=e^{-t\psi(\xi)}},\;\xi\in\mathbb{R}^d,\;t\geq0. \] The main results of the paper are the estimates for the real and the imaginary part of the \(\psi\) in the case when the process \(X\) is \textit{operator semistable} with \({E\in\mathcal{L}(\mathbb{R}^d)}.\) The last means that the distribution \(\mu\) of \(X(1)\) is full and fulfills \[ \mu^c=(c^E\mu)*\varepsilon_u \] for some \({c>1}\) and \({u\in\mathbb{R}^d}\) (for a comprehensive overview see the monograph [\textit{M. M. Meerschaert} and \textit{H.-P. Scheffler}, Limit distributions for sums of independent random vectors. Heavy tails in theory and practice. Chichester: Wiley (2001; Zbl 0990.60003)]). The obtained estimates are used for computing the Hausdorff and the packing measure for the range \(X[0,1]\), for the graph \(\{(t,X(t)):\;t\in[0,1]\}\) as well as for the double point set of \(X.\)