Research Article | Open Access
A. C. Okoroafor, "The Packing Measure of the Trajectory of a One-Dimensional Symmetric Cauchy Process", International Journal of Stochastic Analysis, vol. 2008, Article ID 564601, 7 pages, 2008. https://doi.org/10.1155/2008/564601
The Packing Measure of the Trajectory of a One-Dimensional Symmetric Cauchy Process
Let be a one-dimensional symmetric Cauchy process. We prove that, for any measure function, is zero or infinite, where is the -packing measure of , thus solving a problem posed by Rezakhanlou and Taylor in 1988.
Let be a strictly stable Levy process taking values in (-dimensional Euclidean space) of index , that is, a Markov process with stationary independent increment whose characteristic function is given by
Here, and are points in , is the ordinary inner product in , and . The Levy exponent is of the form where is an arbitrary finite measure on the unit sphere in , not supported on a diametrical plane. If in (1.2) is the uniform distribution on , is called the isotropic stable Levy process with index . In this case, for some . When , must also have the origin as its center of mass, that is, and the resulting process is the symmetric Cauchy process.
If for we have the strictly asymmetric Cauchy process. When we obtain the standard Brownian motion.
We assume that our process has been defined so that the strong Markov property is valid and all sample paths are right continuous and have left limits everywhere.
It is well known that the sample paths of strictly stable Levy processes determine trajectories in that are random fractal sets.
We are interested in the range of the processes, that is, the random set generated by and defined by
The Hausdorff and packing measures serve as useful tools for analyzing fine properties of Levy processes.
The problem of determining the exact Hausdorff measure of the range of those processes for has been completely solved. See, for example, .
The study of the exact packing measure of the range of a stochastic process has a more recent history, starting with the work of Taylor and Tricot .
The packing measure of the trajectory was found in  by Taylor and Tricot for transient Brownian motion. The corresponding problem for the range of strictly stable processes, was solved by Taylor .
For the critical cases, the only known result is due to Le Gall and Taylor . They proved that if is a planar Brownian motion, , is either zero or infinite for any measure function . Hence, the packing measure problem of the symmetric Cauchy process on the line remained open.
The main objective of this paper is to show that for a similar result to that of planner Brownian motion holds for the packing measure of the range of a one-dimensional symmetric Cauchy process with different criteria on .
In this section, we start by recalling the definition and properties of packing measure and packing dimension introduced by Taylor and Tricot .
Let be the class of functions: which are right continuous and monotone increasing with and for which there is a finite constant with
The inequality (2.2) is a weak smoothness condition usually called a doubling property. A function in is often called a measure function: where denotes the closure of the open ball which is centered at and has radius .
A sequence of closed balls satisfying the condition on the right side of (2.3) is called a -packing of .
is a -packing premeasure. The -packing measure on , denoted by , is obtained by defining It is proved in  that is a metric outer measure, and hence every Borel set in is measurable.
We can see that for any , This gives a way to determine the upper bound of . Using the function , gives the fractal index called the packing dimension of .
Lemma 2.1. For a given , there exists a finite constant such that for any Borel measure on with and any Borel set , where is the lower -density of at .
One then uses the sample path to define the random measure known as the occupation measure of the trajectory; denotes the Lebesgue measure.
This gives a Borel measure with , and it is concentrated on and spreads evenly on it.
If then is the sojourn time of in the ball up to the time . Define then by a result in  about one has where is the associated expectation for the process started at 0. Denote by . If , one denotes by .
In , we exhibited a measure function satisfying the following criteria.
Theorem 2.2. Suppose , where is a monotone nondecreasing function and then where is a one-dimensional symmetric Cauchy process.
For any , is also a symmetric Cauchy process on the line since the finite-dimensional distribution of is independent of ; see, for example,  for the strong Markov property of Cauchy processes.
The following corollary is then immediate.
Corollary 2.3. Let , , be a one-dimensional symmetric Cauchy process. Then, for any with probability one, where is as defined in Theorem 2.2.
One will also need an estimate for the small ball probability of the sojourn time , taken from [8, Theorem 3.1].
Lemma 2.4. Suppose , where is a monotone increasing function. For defined in (2.12), then for any fixed constant and , ,
In the next section, we will use the above results and some known techniques to calculate the packing measure of the trajectory of the one-dimensional symmetric Cauchy process.
3. The Measure of the Trajectory
In this section, we proceed to the main result.
Theorem 3.1. Let be a one-dimensional symmetric Cauchy process. If , where is a nondecreasing function, then with probability one, where is the -packing measure of .
Proof. In order to apply the density Lemma
2.1, we have to calculate But by Corollary 2.3, for each fixed with probability one, Then a Fubini argument gives so that if ,
then and a.s. Using an application of the inequality of the
density Lemma 2.1, we have and thus with probability one if .
In order to prove the upper bound, we use density Lemma 2.1 in the other direction, as well as a “bad-point” argument similar to that in .
For each point let denote a semidyadic interval with length whose complement is at distance from a dyadic interval of length which contains .
Let We use the intervals in to replace the balls in (2.3) with length replacing .
This gives a new premeasure comparable to as follows. There exist positive finite constants , such that, for all Borel sets , where For let be the set of “good” points. A Fubini argument tells us that a.s.; then using the density lemma in the other direction, we have Let where is the set of “bad” points.
For , by monotonicity, we have for a positive constant , for infinitely many .
For fixed , we can only get a contribution to from semidyadic intervals of length if the dyadic interval of length is entered by at time and the restarted process leaves the interval of length in less than .
Thus, if is a semidyadic interval of length , then is bad if enters inside dyadic interval of length but spends less than in ; otherwise it is “good”. Any will be in infinitely many such bad .
The probability that is bad given that it is entered is at most by Lemma 2.4.
Let be the number of intervals of length that are entered by the time , and let denote the number of those that are bad; then Leaving out the nonoverlapping requirement, we have, for a positive constant , . Now, by [1, Lemma 4.1], , for a positive constant .
Thus, using (3.16), we have since if for sufficiently small.
It follows that a.s., and from (3.8), a.s. So .
By (3.12), , and therefore if .
This completes the proof.
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Copyright © 2008 A. C. Okoroafor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.