A short proof of the dimension formula for Levy processes

We provide a simple proof of the result on the Hausdorff dimension of the range of a Levy process in a recent paper by Khoshnevisan, Xiao, and Zhong [1]. Let X-t be a Levy process in R-d with the Levy exponent Psi. There was an "open question" which did not garnered lots of attention until a recent paper on multiparameter Levy processes by Khoshnevisan et al. [ 1] showed the simplification of Pruitt's formula in [ 2] as one of the main applications of their long-proof theorems. Khoshnevisan et al. [ 1] obtained: dim(H) X([0, 1]) = sup{alpha < d : integral | y| Re alpha-d(1/1+Psi(y)) dy < infinity} a. s. ( 1.1) where X([0, 1]) = {X-s : s is an element of [0, 1]} with the notation dim(H) for the Hausdorff dimension. The present author is still puzzled by why Pruitt himself did not reach the same conclusion whereas he made some interesting remarks about the difficulty of this issue. We show that Pruitt's elegant proof in [2] also yields (1.1).