Lévy Processes: Capacity and Hausdorff Dimension
نویسنده
چکیده
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 [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 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 ⊂ 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
منابع مشابه
Additive Lévy Processes: Capacity and Hausdorff Dimension
This is a survey on recently-developed potential theory of additive Lévy processes and its applications to fractal geometry of Lévy processes. Additive Lévy processes arise naturally in the studies of the Brownian sheet, intersections of Lévy processes and so on. We first summarize some recent results on the novel connections between an additive Lévy process X in R , and a natural class of ener...
متن کاملDimension Results for Sample Paths of Operator Stable Lévy Processes
Let X = {X(t), t ∈ R+} be an operator stable Lévy process in R with exponent B, where B is an invertible linear operator on R. We determine the Hausdorff dimension and the packing dimension of the range X([0, 1]) in terms of the real parts of the eigenvalues of B. Running Title Dimension of Operator Stable Lévy Processes
متن کاملLevel Sets of Additive Lévy Processes
We provide a probabilistic interpretation of a class of natural capacities on Euclidean space in terms of the level sets of a suitably chosen multiparameter additive Lévy process X . We also present several probabilistic applications of the aforementioned potential-theoretic connections. They include areas such as intersections of Lévy processes and level sets, as well as Hausdorff dimension co...
متن کاملSample Path and Fractal Properties of Lévy Processes
The sample paths of Lévy processes generate various random fractals and many of their properties have been studied since 1960’s [see the survey papers of Fristedt (1974), Taylor (1986) and Xiao (2004)]. The lectures will cover the following topics: Hausdorff dimension and exact Hausdorff measure functions for random fractals determined by Lévy processes; Hausdorff dimension computation using po...
متن کاملHausdorff Dimension of the Contours of Symmetric Additive Lévy Processes
Let X1, . . . , XN denote N independent, symmetric Lévy processes on R. The corresponding additive Lévy process is defined as the following N -parameter random field on R: (0.1) X(t) := X1(t1) + · · ·+ XN (tN ) (t ∈ R+ ). Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set X−1({0}) of X to be non-trivial with positive probability. They also provide boun...
متن کاملAsymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties
This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on Rd . These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy expone...
متن کامل