Packing dimension of the range of a Lévy process

Abstract:

Let $\{X(t)\}_{t\ge 0}$ denote a Lévy process in ${\mathbf {R}}^d$ with exponent $\Psi$. Taylor (1986) proved that the packing dimension of the range $X([0 ,1])$ is given by the index \begin{equation*} {(0.1)}\qquad \qquad \gamma ’ = \sup \left \{\alpha \ge 0: \liminf _{r \to 0^+} \int _0^1 \frac {\mathrm {P} \left \{|X(t)| \le r\right \}}{r^\alpha } dt =0\right \}.\qquad \qquad \end{equation*} We provide an alternative formulation of $\gamma ’$ in terms of the Lévy exponent $\Psi$. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.