Exact Hausdorff Measure Research Articles

The exact hausdorff measure of the zero set of a stable process

The exact hausdorff measure of the zero set of a stable process

The exact Hausdorff measure of the sample path for planar Brownian motion

It has long been known (see Lévy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to tα is infinite for 0 < α < 2 with probability 1 so that the dimension of the path set is known to be 2. If one considers the initial part of the sample path Cω = C(1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ(t) such that, with probability 1,for suitable positive constants c1, c2. The corresponding problem for paths in k-space (k ≥ 3) has been solved. In this case, if φ1(t) = t2 log log t−1, Lévy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ1(t) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ2(t) = t2 log t−1, and at that time we thought that (1) might be true with ψ(t) = φ2(t). Recently Ray (5) has obtained the lower bound in (1)withThe purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.

Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion

Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion

First Passage times and Sojourn Times for Brownian Motion in Space and the Exact Hausdorff Measure of the Sample Path

First Passage times and Sojourn Times for Brownian Motion in Space and the Exact Hausdorff Measure of the Sample Path

First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the definition of 4-measure. In the past the only successful method has been to use the connection between Hausdorff measures and generalized capacities. The first result of this kind appears in [16] where it is proved that for k >2, the Hausdorff measure with respect to ta is infinite for all a 3 we obtain a law of the iterated logarithm for the total time Tk(a, co) spent by the path X in a sphere of radius a as a-* 0+. We also determine the local behavior of the first passage time Pk(a, co) out of a sphere of radius a for k > 1. In order to obtain the required asymptotic results we required good estimates of the distribution functions for the random variables Tk(a, co), Pk(a, c). We use the method developed by Mark Kac to compute these dis-