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Lévy processes: capacity and Hausdorff dimension. (English) Zbl 1072.60040

The authors establish novel connections between an arbitrary Lévy process \(X\) in \(\mathbb{R}^{d}\), \(d\geq 1\), and a new class of energy forms and their corresponding capacities by using the recently developed multiparameter theory of additive Lévy processes. Then they apply these connections to solve two long-standing problems in the Lévy process theory. First, the authors compute the Hausdorff dimension of the image \(X(G) \) of a (nonrandom) Borel set \(G\subset \mathbb{R}_{+}.\) This completes previous works of S. J. Taylor (1953), H. P. McKean jun. (1955), R. M. Blumenthal and R. K. Getoor (1960, 1961), W. E. Pruitt (1969), W. E. Pruitt and S. J. Taylor (1969), P. W. Millar (1971), J. Hawkes (1971, 1978, 1998), W. J. Hendricks (1972, 1973), J.-P. Kahane (1983), P. Becker-Kern, M. M. Meerschaert and H.-P. Scheffler (2003), and D. Khoshnevisan, Y. Xiao and Y. Zhong (2003), where \(\dim X(G) \) is computed under various conditions on \(G\), \(X\) or both. Second, the authors solve a 1983 problem of J.-P. Kahane asking for a necessary and sufficient analytic condition on any two disjoint Borel sets \(F,G\subset \mathbb{R}_{+}\) such that \(X(F) \cap X(G) \neq \emptyset\) when \(X\) is an isotropic stable process. They present a solution for Lévy processes \(X\) in \( \mathbb{R}^{d}\) such that the distribution of \(X(t) \) is equivalent to Lebesgue measure on \(\mathbb{R}^{d}\) for all \(t>0.\) Prior to the present paper, the case of a Brownian motion \(X\) only was understood [D. Khoshnevisan (1999)]. Finally, the authors compute the Hausdorff dimension and capacity of the preimage \(X^{-1}(F) \) of a Borel set \(F\subset \mathbb{R}^{d}\) under mild assumptions on \(X\). This completes the work of J. Hawkes (1998) who took up the special case where \(X\) is a subordinator.

MSC:

60G51 Processes with independent increments; Lévy processes
28A80 Fractals
60G17 Sample path properties
60J25 Continuous-time Markov processes on general state spaces
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