Packing dimension of the range of a Lévy process

Abstract

Let <!-- MATH $\{X(t)\}_{t\ge 0}$ --> denote a Lévy process in <!-- MATH ${\mathbf{R}}^d$ --> with exponent . Taylor (1986) proved that the packing dimension of the range <!-- MATH $X([0\,,1])$ --> is given by the index <!-- MATH \begin{displaymath} {(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{displaymath} -->