ar X iv : m at h / 04 06 44 4 v 1 [ m at h . PR ] 2 2 Ju n 20 04 LÉVY PROCESSES : CAPACITY AND HAUSDORFF DIMENSION

We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in R, and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Lévy processes. First, we compute the Hausdorff dimension of the image X(G) of a nonrandom linear Borel set G ⊂ R+, where X is an arbitrary Lévy process in R. 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 a necessary and sufficient analytic condition on any two disjoint Borel sets F,G ⊂ R+ such that with positive probability, X(F )∩X(G) is nonempty? 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 R for all t > 0. As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage X(F ) of a nonrandom Borel set F ⊂ R 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. Date: November 21, 2003. 2000 Mathematics Subject Classification. Primary 60J25, 28A80; Secondary 60G51, 60G17.