First, we compute the Hausdorff dimension of the image X(G) of a nonrandom linear Borel set G in R+, where X is an arbitrary Lévy process in Rd. Our work completes the various earlier efforts of Taylor (1953), McKean (1955), Blumenthal and Getoor (1960; 1961), Millar (1971), Pruitt (1969), Pruitt and Taylor (1969), Hawkes (1971; 1978; 1998), Hendricks (1972; 1973), Kahane (1983; 1985b), Becker-Kern, Meerschaert, and Scheffler (2003), and Khoshnevisan, Xiao, and Zhong (2003a), where dimX(G) is computed under various conditions on G, X, or both.
We next solve the following problem (Kahane, 1983): When X is an isotropic stable process, what is an analytic necessary and sufficient condition on any two disjoint Borel sets F,G in Rd such that with positive probability X(F) intersects X(G)? Prior to this article, this was understood only in the case that X is Brownian motion (Khoshnevisan, 1999). Here, we present a solution to Kahane's problem for an arbitrary Lévy process X provided the distribution of X(t) is mutually absolutely continuous with respect to the Lebesgue measure on Rd for all t>0.
As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage X-1(F) of a nonrandom Borel set F in Rd under very mild conditions on the process X. This completes the work of Hawkes (1998) that covers the special case where X is a subordinator.
Keywords. Lévy and additive Lévy process, capacity, Hausdorff dimension.
AMS Classification (2000) Primary. 60J25, 28A80. Secondary. 60G51, 60G17.
Support. Research supported in part by a grant from the National Science Foundation.
Pre/E-Prints. This paper is available in
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Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |
Yimin Xiao Department of Statistics and Probability A-413 Wells Halls Michigan State University East Lansing MI 48824 xiao@stt.msu.edu |
Last Update: June 18, 2003
© 2003 - Davar Khoshnevisan and Yimin Xiao