Levy processes: Capacity and Hausdorff dimension

Details Levy processes: Capacity and Hausdorff dimension Levy processes: Capacity and Hausdorff dimension

2004-06-22

arxiv.org/abs/math/0406444v2

Annals of Probability 2005, Vol. 33, No. 3, 841-878

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000001026 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000001026

We use the recently-developed multiparameter theory of additive Levy processes to establish novel connections between an arbitrary Levy process $X$ in $\mathbf{R}^d$, 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 $G\subset \mathbf{R}_+$, where $X$ is an arbitrary Levy process in $\mathbf{R}^d$. 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.

Affiliated with

Also associated with

No associations

Rating

Levy processes: Capacity and Hausdorff dimension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Levy processes: Capacity and Hausdorff dimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Levy processes: Capacity and Hausdorff dimension will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-669911