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The Hausdorff α-dimensional measure of Brownian paths in n-space

Published online by Cambridge University Press:  24 October 2008

S. J. Taylor
Affiliation:
Peterhouse Cambridge

Extract

Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3. However, if we apply the method indicated by Hausdorff (4) for constructing a linear set having measure 1 with respect to a given measure function h(x), the Cantor-type set we obtain when h(x) = xα log log 1/x, where 0 < α < 1, is easily seen to have zero α-capacity but infinite α-measure; and similar methods apply for other values of α. Thus the result mentioned above does not imply that Brownian paths in n-space (n ≥ 3) have zero 2-measure. The other relevant result is due to Lévy (6), who showed that Brownian paths in the plane have zero Lebesgue measure (and therefore zero Hausdorff 2-measure) with probability 1. However, his method of proof cannot be extended to deal with Brownian paths in n-space.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

REFERENCES

(1)Cameron, R. H. and Martin, W. T.An expression for the solution of non-linear integral equations. Amer. J. Math. 66 (1944), 281–98.CrossRefGoogle Scholar
(2)Doob, J. L.Probability in function space. Bull. Amer. math. Soc. 53 (1947), 1530.CrossRefGoogle Scholar
(3)Dvoretsky, A., Erdös, P. and Kakutani, S.Brownian motion in n-space. Acta Scientiarum mathematicarum Univ. Szeged, 12B (1950), 7581.Google Scholar
(4)Hausdorff, F.Dimension und äusseres Mass. Math. Ann. 79 (1918), 157–79.CrossRefGoogle Scholar
(5)Lévy, P.Sur certains processus stochastiques homogènes. Compos. math. 7 (1940), 283339.Google Scholar
(6)Lévy, P.Le mouvement Brownien plan. Amer. J. Math. 62 (1940), 487550.CrossRefGoogle Scholar
(7)Paley, R. E. A. C. and Wiener, N.Fourier transforms in the complex domain (Colloq. Publ. Amer. math. Soc. no. 19, 1934).Google Scholar
(8)Taylor, S. J.On Cartesian product sets. J. Lond. math. Soc. 27 (1952), 295304.CrossRefGoogle Scholar
(9)Wiener, N.Differential space. J. Math. Phys. 2 (1923), 131–74.CrossRefGoogle Scholar
(10)Wiener, N.Generalized harmonic analysis. Acta math., Stockh., 55 (1930), 117258.CrossRefGoogle Scholar