Packing Dimension of the Range of a Lévy Process

Davar Khoshnevisan and Yimin Xiao

Abstract. Let {X(t)}t≥0 denote a Lévy process in Rd with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X([0,1]) is given by the index

γ' = sup{α≥0: liminfr↓0 r01 P {|X(t)|≤r} 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.

Keywords. Lévy processes, operator stable Lévy processes, packing dimension, Hausdorff dimension.

AMS Classification (2000). 60J30, 60G17, 28A80.

Support. Research supported in part by grants from the U.S. National Science Foundation.

Pre/E-Prints. This paper is available in

Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
Yimin Xiao
Department of Statistics and Probability
A-413 Wells Hall
Michigan State University
East Lansing, MI 48824, U.S.A.

Last Update: March 7, 2007
© 2002 - Davar Khoshnevisan and Yimin Xiao