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Lévy Processes

其他書名	Capacity and Hausdorff Dimension
出版	SSRN, 2018
URL	http://books.google.com.hk/books?id=kJr9zgEACAAJ&hl=&source=gbs_api

註釋We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process 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 () of a non-random linear Borel set ⊂ R, where 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 dim () is computed under various conditions on , , or both.We next solve the following problem (Kahane, 1983): ⊂ R ( ) ∩ () Prior to this article, this was understood only in the case that is Brownian motion (Khoshnevisan, 1999). Here, we present a solution to Kahane's problem for an arbitrary Lévy process provided the distribution of () is mutually absolutely continuous with respect to the Lebesgue measure on R for all > 0.As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage () of a nonrandom Borel set ∈ R under very mild conditions on the process . This completes the work of Hawkes (1998) that covers the special case where is a subordinator.