Lévy processes: Capacity and Hausdorff dimension

Abstract

We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in R^d, 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^d. Our work completes the various earlier efforts of Taylor [Proc. Cambridge Phil. Soc. 49 (1953) 31–39], McKean [Duke Math. J. 22 (1955) 229–234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370–375, J. Math. Mech. 10 (1961) 493–516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53–73], Pruitt [J. Math. Mech. 19 (1969) 371–378], Pruitt and Taylor [Z. Wahrsch. Verw. Gebiete 12 (1969) 267–289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90–102, J. London Math. Soc. (2) 17 (1978) 567–576, Probab. Theory Related Fields 112 (1998) 1–11], Hendricks [Ann. Math. Stat. 43 (1972) 690–694, Ann. Probab. 1 (1973) 849–853], Kahane [Publ. Math. Orsay (83-02) (1983) 74–105, Recent Progress in Fourier Analysis (1985b) 65–121], Becker-Kern, Meerschaert and Scheffler [Monatsh. Math. 14 (2003) 91–101] and Khoshnevisan, Xiao and Zhong [Ann. Probab. 31 (2003a) 1097–1141], where dim X(G) is computed under various conditions on G, X or both.

We next solve the following problem [Kahane (1983) Publ. Math. Orsay (83-02) 74–105]: 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 a Brownian motion [Khoshnevisan (1999) Trans. Amer. Math. Soc. 351 2607–2622]. 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^d 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⊂R^d under very mild conditions on the process X. This completes the work of Hawkes [Probab. Theory Related Fields 112 (1998) 1–11] that covers the special case where X is a subordinator.