Journal of Applied Mathematics and Stochastic Analysis

VolumeΒ 2008Β (2008), Article IDΒ 564601, 7 pages

http://dx.doi.org/10.1155/2008/564601

## The Packing Measure of the Trajectory of a One-Dimensional Symmetric Cauchy Process

Department of Mathematics, Abia State University, 440001 Uturu, Nigeria

Received 3 August 2007; Revised 27 May 2008; Accepted 12 August 2008

Academic Editor: MohsenΒ Pourahmadi

Copyright Β© 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.

#### Abstract

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.

#### 1. Introduction

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, [1].

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 [2].

The packing measure of the trajectory was found in [2] by Taylor and Tricot for transient Brownian motion. The corresponding problem for the range of strictly stable processes, was solved by Taylor [3].

Further results on the asymmetric Cauchy process and subordinators have been established by Rezakhanlou and Taylor [4] and Fristedt and Taylor [5], respectively.

For the critical cases, the only known result is due to Le Gall and Taylor [6]. 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 .

#### 2. Preliminaries

In this section, we start by recalling the definition and properties of packing measure and packing dimension introduced by Taylor and Tricot [2].

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 *Β *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 [2] 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 .

In order to calculate the packing measure, we will use the following density theorem of Taylor and Tricot [2], which we will call Lemma 2.1.

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 [7] about one has where is the associated expectation for the process started at 0. Denote by . If , one denotes by .

In [8], 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, [1] 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 βbad-pointβ argument
similar to that in [3].

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 βgoodβ 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 βbadβ 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 βgoodβ. 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.

#### References

- Y. Xiao, βRandom fractals and Markov processes,β in
*Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2*, M. L. Lapidus and M. van Frankenhuijsen, Eds., vol. 72 of*Proceedings of Symposia in Pure Mathematics*, pp. 261β338, American Mathematical Society, Providence, RI, USA, 2004. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. J. Taylor and C. Tricot, βPacking measure, and its evaluation for a Brownian path,β
*Transactions of the American Mathematical Society*, vol. 288, no. 2, pp. 679β699, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. J. Taylor, βThe use of packing measure in the analysis of random sets,β in
*Stochastic Processes and Their Applications*, vol. 1203 of*Lecture Notes in Mathematics*, pp. 214β222, Springer, Berlin, Germany, 1986. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - F. Rezakhanlou and S. J. Taylor, βThe packing measure of the graph of a stable process,β
*Astérisque*, no. 157-158, pp. 341β362, 1988. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - B. E. Fristedt and S. J. Taylor, βThe packing measure of a general subordinator,β
*Probability Theory and Related Fields*, vol. 92, no. 4, pp. 493β510, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - J.-F. Le Gall and S. J. Taylor, βThe packing measure of planar Brownian motion,β in
*Seminar on Stochastic Processes*, pp. 130β148, Birkhäuser, Boston, Mass, USA, 1986. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - R. Bañuelos and T. Kulczycki, βThe Cauchy process and the Steklov problem,β
*Journal of Functional Analysis*, vol. 211, no. 2, pp. 355β423, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. C. Okoroafor, βSome local asymptotic laws for the Cauchy process on the line,β
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 2007, Article ID 81934, 9 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet