Packing Dimension of the Range of a Lévy Process

Let {X(t)}t≥0 denote a Lévy process in R with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X([0 , 1]) is given by the index (0.1) γ′ = sup { α ≥ 0 : lim inf r→0+ ∫ 1 0 P {|X(t)| ≤ r} rα dt = 0 } . We provide an alternative formulation of γ′ in terms of the Lévy exponent Ψ. 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.