Hostname: page-component-7b9c58cd5d-dkgms Total loading time: 0 Render date: 2025-03-15T11:18:15.025Z Has data issue: false hasContentIssue false
  • English
  • Français

The α-dimensional measure of the graph and set of zeros of a Brownian path

Published online by Cambridge University Press:  24 October 2008

S. J. Taylor
S. J. Taylor
Affiliation:
PeterhouseCambridge
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent joint paper (1) with Prof. Besicovitch we announced the conjecture that for almost all one-dimensional Brownian paths, the set of zeros has dimensional number ½, and zero A½-measure. It is the purpose of this paper to give a proof of this result. In doing so we consider the graph C(ω) of a Brownian path ω as a point set in the plane, and prove that, with probability 1, C(ω) has dimensional number ¾ and zero Λ¾-measure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 265 - 274
Copyright
Copyright © Cambridge Philosophical Society 1955

References

REFERENCES

(1)Besicovitch, A. S. and Taylor, S. J.On the complementary intervals of a linear closed set of zero Lebesgue measure. J. Lond. math. Soc. 29 (1954), 449–59.CrossRefGoogle Scholar
(2)Besicovitch, A. S. and Ursell, H. D.Sets of fractional dimensions, V: On dimensional numbers of some continuous curves. J. Lond. math. Soc. 12 (1937), 18–25.CrossRefGoogle Scholar
(3)Fekete, M.Über den transfiniten Durchmesser ebener Punktmengen. I. Math. Z. 32 (1930), 108–14.CrossRefGoogle Scholar
(4)Frostman, O.Potentiel d'équilibre et capacité des ensembles, avec quelques applications à la théorie des fonctions. Medd. Lunds Univ. mat. Semin. 3 (1935).Google Scholar
(5)Kametani, S.On Hausdorff's measures and generalised capacities with some of their applications to the theory of functions. Jap. J. Math. 19 (1946), 217–57.CrossRefGoogle Scholar
(6)LÉvy, P.Sur certains processus stochastiques homogènes. Compos. math. 7 (1940), 283–339.Google Scholar
(7)LÉvy, P.Processus stochastiques et mouvements browniens (Paris, 1948).Google Scholar
(8)LÉVY, P.La mesure de Hausdorff de la courbe du mouvement brownien. G. Inst. ital. Attuari, 16 (1953), 1–37.Google Scholar
(9)Marstrand, J. M.The dimension of Cartesian product sets. Proc. Camb. phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
(10)Paley, R. E. A. C. and Wiener, N.Fourier transforms in the complex domain (Colloq. Publ. Amer. math. Soc. no. 19, 1934).Google Scholar
(11)Taylor, S. J.The Hausdorff α-dimensional measure of Brownian paths in n–space. Proc. Camb. phil. Soc. 49 (1953), 31–9.CrossRefGoogle Scholar
(12)Weyl, H.Über die Gleichwerteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–54.CrossRefGoogle Scholar