899 research outputs found
Brownian motion and thermal capacity
Let denote -dimensional Brownian motion. We find an explicit formula
for the essential supremum of Hausdorff dimension of , where
and are arbitrary nonrandom
compact sets. Our formula is related intimately to the thermal capacity of
Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when
, our formula can be described in terms of the Hausdorff dimension of
, where is viewed as a subspace of space time.Comment: Published in at http://dx.doi.org/10.1214/14-AOP910 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Levy processes: Capacity and Hausdorff dimension
We use the recently-developed multiparameter theory of additive Levy
processes to establish novel connections between an arbitrary Levy process
in , 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 of a nonrandom linear Borel set
, where is an arbitrary Levy process in
. 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 is computed under
various conditions on , or both.Comment: Published at http://dx.doi.org/10.1214/009117904000001026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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