Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  101.11204
Autor:  Erdös, Pál; Taylor, S.J.
Title:  On the Hausdorff measure of Brownian paths in the plane (In English)
Source:  Proc. Camb. Philos. Soc. 57, 209-222 (1961).
Review:  Let us denote by \Omega the set of all brownian plane paths z(t,\omega) = (z(t,\omega),y(t,\omega)) where \omega is a random point and 0 < t < oo. One of the two authors (S.J.Taylor, Zbl 050.05803) constructed a probabilistical device {\Omega F,\mu} for the space of Brownian motion.
Paul Lévy has proved (Zbl 024.13906) that the Lebesgue plane measure of the set L(0, oo; \omega) [where L(a,b; \omega) = {z(t,\omega) | 0 \leq a < t < l \leq oo}] is -- with probability one -- equal to null. In the present paper the authors prove Lévy's conjecture i. e. that, in contrast to the occurrences in the multidimensional case, the measure of the set L(0,1; \omega) in the twodimensional space is finite, with respect to function -x2 log x. The method employed in the demonstration uses the connexion between the Hausdorff-measure and the generalized capacity, that was pointed out by S.Kametani [Jap. J. Math. 19, 217-257 (1946; Zbl 061.22704)].
Reviewer:  O.Onicescu
Classif.:  * 60J65 Brownian motion
                   28A78 Hausdorff measures
Index Words:  probability theory

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