Lévy processes: Capacity and Hausdorff dimension

By Davar Khoshnevisan, Yimin Xiao. 2004

60 citations recorded by the source corpus.

We use the recently-developed multiparameter theory of additive Levy processes to establish novel connections between an arbitrary Levy 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 Levy processes. First, we compute the Hausdorff dimension of the image X(G) of a nonrandom linear Borel set G C R + , where X is an arbitrary Levy 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, Probah. 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 dimX(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 C R + such that with positive probability, X(F) n 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 Levy 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.

Subfields: Finance, Management Information Systems, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/0c0e74cd964ce407b7bb8218551e161bd4d90867

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Citations in the corpus

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields (2009, methodology): “The analogous problem for Lévy processes can be solved by applying potential theory of Lévy processes and a subordination technique (see [7, 12] for more details).”
Sample Path Properties of Anisotropic Gaussian Random Fields (2009, background): “Recently, Biermé, Lacaux and Xiao (2007) determined the Hausdorff dimension of X−1(F ).”
Harmonic analysis of additive Lévy processes (2007, background): “Also, we showed in our earlier works [67,68,70] that such a theorem has a large number of consequences, many of them in the classical theory of Lévy processes itself.”