International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2008 / Article

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Volume 2008 |Article ID 564601 | https://doi.org/10.1155/2008/564601

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

Academic Editor: Mohsen Pourahmadi
Received03 Aug 2007
Revised27 May 2008
Accepted12 Aug 2008
Published28 Sep 2008

Abstract

Let 𝑋𝑡={𝑋(𝑡),𝑡≥0} be a one-dimensional symmetric Cauchy process. We prove that, for any measure function, 𝜑,𝜑−𝑝(𝑋[0,𝜏]) is zero or infinite, where 𝜑−𝑝(𝐸) is the 𝜑-packing measure of 𝐸, thus solving a problem posed by Rezakhanlou and Taylor in 1988.

1. Introduction

Let 𝑋𝑡={𝑋(𝑡),𝑡≥0} be a strictly stable Levy process taking values in 𝑅𝑛 (𝑛-dimensional Euclidean space) of index 𝛼∈(0,2], that is, a Markov process with stationary independent increment whose characteristic function is given by 𝐸𝑒𝑖𝑢,𝑋𝑡=𝑒−𝑡𝜓𝛼𝑢.(1.1)

Here, 𝑢 and 𝑋𝑡 are points in 𝑅𝑛, (𝑢,𝑥) is the ordinary inner product in 𝑅𝑛, and ‖𝑥‖2=(𝑥,𝑥). The Levy exponent 𝜓𝛼(𝑢) is of the form 𝜓𝛼𝑢=||𝑢||𝛼𝑆𝑛𝑤𝛼𝜇𝑢,𝑦𝑑𝑦,(1.2)where 𝑤𝛼=𝑢,𝑦1−𝑖sgn𝑢,𝑦tan𝜋𝛼2|||𝑢‖‖𝑢‖‖|||,𝑦𝛼𝑤if𝛼≠1,1=|||𝑢𝑢,𝑦‖‖𝑢‖‖|||+,𝑦2𝑖𝜋||||||.𝑢,𝑦log𝑢,𝑦(1.3)𝜇(𝑑𝑦) is an arbitrary finite measure on the unit sphere 𝑆𝑛 in 𝑅𝑛, not supported on a diametrical plane. If in (1.2) 𝜇 is the uniform distribution on 𝑆𝑛, 𝑋𝑡 is called the isotropic stable Levy process with index 𝛼. In this case, 𝜓𝛼(𝑢)=𝜆|𝑢|𝛼 for some 𝜆>0. When 𝛼=1, 𝜇 must also have the origin as its center of mass, that is, 𝑆𝑛𝑦𝜇𝑑𝑦=0,(1.4)and the resulting process is the symmetric Cauchy process.

If ∫𝑆𝑛𝑦𝜇(𝑑𝑦)≠0 for 𝛼=1, we have the strictly asymmetric Cauchy process. When 𝛼=2, 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 𝑋𝑡 of strictly stable Levy processes determine trajectories in 𝑅𝑛 that are random fractal sets.

We are interested in the range of the processes, that is, the random set 𝑅𝜏 generated by 𝑋𝑡 and defined by 𝑅𝜏==𝑋0,𝜏𝑥∈𝑅𝑛𝑡∶𝑥=𝑋forsome𝑡∈0,𝜏.(1.5)

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 𝛼∈(0,2] 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, 𝛼<𝑛, 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, 𝛼=𝑛, the only known result is due to Le Gall and Taylor [6]. They proved that if 𝑋(𝑡) is a planar Brownian motion, 𝛼=𝑛=2, 𝜑−𝑝[𝑋([0,𝑡])] is either zero or infinite for any measure function 𝜑. 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 𝛼=𝑛=1, 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 𝜑.

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 Φ be the class of functions: →𝜑∶0,𝛿0,∞(2.1) which are right continuous and monotone increasing with 𝜑(0+)=0 and for which there is a finite constant 𝑘>0 with 𝜑2𝑠𝜑𝑠𝛿≤𝑘for0<𝑠<2.(2.2)

The inequality (2.2) is a weak smoothness condition usually called a doubling property. A function 𝜑 in Φ is often called a measure function: 𝐸𝜑−𝑃=limsup𝜀→0𝑖𝜑2𝑟𝑖∶𝐵𝑥𝑖,𝑟𝑖aredisjoint,𝑥𝑖∈𝐸,𝑟𝑖<𝜀,(2.3) where 𝐵(𝑥𝑖,𝑟𝑖) denotes the closure of the open ball 𝐵(𝑥𝑖,𝑟𝑖) which is  centered at 𝑥 and has radius 𝑟.

A sequence of closed balls satisfying the condition on the right side of (2.3) is called a 𝜀-packing of 𝐸.

𝜑−𝑃 is a 𝜑-packing premeasure. The 𝜑-packing measure on 𝑅𝑛, denoted by 𝜑−𝑝, is obtained by defining 𝐸𝜑−𝑝=inf𝑛𝐸𝜑−𝑃∶𝐸⊆𝑛𝐸𝑛.(2.4)It is proved in [2] that 𝜑−𝑝(𝐸) is a metric outer measure, and hence every Borel set in 𝑅𝑛 is 𝜑−𝑝 measurable.

We can see that for any 𝐸⊂𝑅𝑛, 𝐸𝐸𝜑−𝑝≤𝜑−𝑃.(2.5)This gives a way to determine the upper bound of 𝜑−𝑝(𝐸). Using the function 𝜑(𝑠)=𝑠𝛼, 𝛼>0 gives the fractal index dim𝑝𝐸=inf𝛼>0∶𝑠𝛼𝐸−𝑝=0=sup𝛼>0∶𝑠𝛼𝐸−𝑝=∞,(2.6)called the packing dimension of 𝐸.

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 𝜑∈Φ, there exists a finite constant 𝑘>0 such that for any Borel measure 𝜇 on 𝑅𝑛 with 0<‖𝜇‖=𝜇(𝑅𝑛)<∞ and any Borel set 𝐸⊆𝑅𝑛, 𝑘−1𝜇𝐸inf𝑥∈𝐸𝐷𝜑−𝜇𝑥−1𝐸‖‖𝜇‖‖≤𝜑−𝑝≤𝑘sup𝑥∈𝐸𝐷𝜑−𝜇𝑥−1,(2.7)where 𝐷𝜑−𝜇𝑥=liminf𝑟↓0𝜇𝐵𝑥,𝑟𝜑2𝑟(2.8) is the lower 𝜑-density of 𝜇 at 𝑥.

One then uses the sample path 𝑋𝑡 to define the random measure 𝜇𝐸=|||𝑡|||𝑡∈0,𝜏∶𝑋∈𝐸(2.9) known as the occupation measure of the trajectory; |⋅| denotes the Lebesgue measure.

This gives a Borel measure with 𝜇(𝑅)=‖𝑢‖=𝜏, and it is concentrated on 𝑋[0,𝜏] and spreads evenly on it.

If 𝑡𝑥=𝑋0,0<𝑡0<𝜏,(2.10)then 𝜇𝐵=𝑥,𝑟𝜏0𝐼𝐵𝑥,𝑟𝑋𝑡𝑑𝑡=𝑇𝑥,𝑟(2.11) is the sojourn time of 𝑋𝑡 in the ball 𝐵(𝑥,𝑟) up to the time 𝜏. Define 𝜏=inf{𝑡>0∶|𝑥(𝑡)|>1}; then by a result in [7] about 𝜏 one has 𝐸0𝜏=1, where 𝐸0 is the associated expectation for the process started at 0. Denote ∫∞0 by ∫0+. If 𝑥=0, one denotes 𝑇(𝑥,𝑟) by 𝑇(𝑟).

In [8], we exhibited a measure function 𝜑 satisfying the following criteria.

Theorem 2.2. Suppose 𝜑=ğ‘Ÿâ„Ž(𝑟), where ℎ(𝑟) is a monotone nondecreasing function and 𝑇𝑟=𝜏0𝐼𝐵0,𝑟𝑋𝑡𝑑𝑡;(2.12)then liminf𝑟↓0𝑇𝑟𝜑𝑟=⎧⎪⎨⎪⎩0if0+â„Žî€·ğ‘ î€¸î€·î€¸ğ‘ ln1/𝑠=∞,∞otherwise,(2.13) where 𝑋𝑡 is a one-dimensional symmetric Cauchy process.

For any 𝑡0≥0, 𝑋(𝑡+𝑡0)−𝑋(𝑡0) is also a symmetric Cauchy process on the line since the finite-dimensional distribution of 𝑋(𝑡+𝑡0)−𝑋(𝑡0) is independent of 𝑡0; see, for example, [1] for the strong Markov property of Cauchy processes.

The following corollary is then immediate.

Corollary 2.3. Let 𝑋𝑡, 𝑡≥0, be a one-dimensional symmetric Cauchy process. Then, for any 𝑡0≥0 with probability one, liminf𝑟↓0𝑇𝑋𝑡0,𝑟𝜑𝑟=⎧⎪⎨⎪⎩0if0+â„Žî€·ğ‘ î€¸î€·î€¸ğ‘ ln1/𝑠=∞,∞otherwise,(2.14) where 𝜑 is as defined in Theorem 2.2.

One will also need an estimate for the small ball probability of the sojourn time 𝑇(𝑟), taken from [8, Theorem 3.1].

Lemma 2.4. Suppose 𝜑(𝑟)=ğ‘Ÿâ„Ž(𝑟), where ℎ(𝑟) is a monotone increasing function. For 𝑇 defined in (2.12), then for any fixed constant 𝑐1 and ğ‘Žğ‘˜=𝜌−𝑘, 𝜌>1, ğ‘ƒî‚†ğ‘‡î‚€ğ‘Žğ‘˜+1<𝑐1ğœ‘î‚€ğ‘Žğ‘˜î‚î‚‡â‰¤ğ‘2â„Žî‚€ğ‘Žğ‘˜î‚ğ‘˜.(2.15)

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 𝑋(𝑡)={𝑋(𝑡):𝑡≥0} be a one-dimensional symmetric Cauchy process. If 𝜑(𝑟)=ğ‘Ÿâ„Ž(𝑟), where ℎ is a nondecreasing function, then with probability one, 𝑋=âŽ§âŽªâŽ¨âŽªâŽ©î€œğœ‘âˆ’ğ‘0,𝜏0if0+â„Žî€·ğ‘ î€¸î€·î€¸ğ‘ ln1/𝑠<∞,∞otherwise,(3.1) where 𝜑−𝑝(𝑋([0,𝜏])) is the 𝜑-packing measure of 𝑋([0,𝜏]).

Proof. In order to apply the density Lemma 2.1, we have to calculate liminf𝑟↓0𝜇𝐵𝑥,𝑟𝜑2𝑟.(3.2)But by Corollary 2.3, for each fixed 𝑡0∈(0,𝜏) with probability one, liminf𝑟↓0𝜇𝐵𝑋𝑡0,𝑟𝜑𝑟=liminf𝑟↓0𝑇𝑋𝑡0,𝑟𝜑𝑟=0if0+â„Žî€·ğ‘ î€¸ğ‘ ln(1/𝑠)=∞.(3.3)Then a Fubini argument gives |||𝑡∈0,𝜏∶liminf𝑟↓0𝜇𝐵𝑋𝑡,𝑟𝜑𝑟|||=0a.s.=𝜏<∞(3.4) so that if 𝐸={𝑋(𝑡0):𝑡0∈(0,𝜏)}, then 𝐸⊆𝑋([0,𝜏]) and 𝜇(𝐸)=𝜏<∞ a.s. Using an application of the inequality of the density Lemma 2.1, we have 𝐸𝜑−𝑝=∞,(3.5)and thus 𝜑−𝑝𝑋([0,𝑡])=∞ with probability one if ∫0+(ℎ(𝑠)/𝑠ln(1/𝑠))=∞.
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 [3].
For each point 𝑥∈𝑅, let 𝑉𝑘(𝑥) denote a semidyadic interval with length 2−𝑘 whose complement is at distance 2−𝑘−2 from a dyadic interval of length 2−𝑘−2 which contains 𝑥.
Let Γ𝐸=𝑉𝑘𝑥∶𝑘=1,2,…,𝑥∈𝐸.(3.6)We use the intervals in Γ𝐸 to replace the balls 𝐵(𝑥,𝑟) in (2.3) with length 2−𝑘 replacing 2𝑟=diam𝐵(𝑥,𝑟).
This gives a new premeasure 𝜑−𝑃𝑥𝑥(𝐸) comparable to 𝜑−𝑃 as follows. There exist positive finite constants 𝑘1, 𝑘2 such that, for all Borel sets 𝐸⊂𝑅, 𝑘1𝜑−𝑃𝑥𝑥𝐸𝐸≤𝜑−𝑃≤𝑘2𝜑−𝑃𝑥𝑥𝐸,(3.7) where 𝜑−𝑝𝑥𝑥𝐸=inf𝑖𝜑−𝑃𝑥𝑥𝐸𝑖∶𝐸⊂∪𝐸𝑖.(3.8)For ∞0+â„Žî€·ğ‘ î€¸ğ‘ ln(1/𝑠)<∞,(3.9)let 𝑡𝐺=0∈𝜇𝐵𝑋𝑡0,𝜏∶liminf0,𝑟𝜑2𝑟=∞(3.10) be the set of “good” points. A Fubini argument tells us that |𝐺|=𝜏<∞ a.s.; then using the density lemma in the other direction, we have 𝑋𝐺𝜑−𝑃=0.(3.11)Let 0,𝜏⧵𝐺=âˆžîšğ‘–=1𝐺𝑗,(3.12)where 𝐺𝑗=𝜇𝐵𝑋𝑡𝑡∈0,𝜏∶liminf,𝑟𝜑2𝑟≤𝑗(3.13) is the set of “bad” points.
For 𝑡∈𝐺𝑗, by monotonicity, we have for a positive constant 𝑐, 𝜇𝐵𝑋𝑡,2−𝑘2≤𝑐𝑗𝜑−𝑘,(3.14)for infinitely many 𝑘.
For fixed 𝑗, we can only get a contribution to 𝜑−𝑃𝑥𝑥(𝑋(𝐺𝑗)) from semidyadic intervals of length 2−𝑘 if the dyadic interval of length 2−𝑘−2 is entered by 𝑋(𝑡) at time 𝑡≤𝜏 and the restarted process leaves the interval of length 2−𝑘−2 in less than 𝑗𝜑(2−𝑘).
Thus, if 𝑆𝑘 is a semidyadic interval of length 2−𝑘, then 𝑆𝑘 is bad if 𝑋(𝑠) enters inside dyadic interval of length 2−𝑘−2 but spends less than 𝑗𝜑(2−𝑘) in 𝑆𝑘; otherwise it is “good”. Any 𝑡∈𝐺𝑗 will be in infinitely many such bad 𝑆𝑘.
The probability that 𝑆𝑘 is bad given that it is entered is at most 𝑃𝑇2−𝑘2≤𝑐𝑗𝜑−𝑘≤2î‚î‚‡ğ‘â„Žâˆ’ğ‘˜î‚ğ‘˜,(3.15) by Lemma 2.4.
Let 𝑁𝑘(𝜏) be the number of intervals of length 2−𝑘 that are entered by the time 𝜏, and let 𝐵𝑘(𝜏) denote the number of those that are bad; then ğ¸ğµğ‘˜î€·ğœî€¸â‰¤ğ¸ğ‘ğ‘˜î€·ğœî€¸â„Žî‚€2−𝑘𝑘.(3.16)Leaving out the nonoverlapping requirement, we have, for a positive constant 𝑐3, 𝐸𝜑−𝑝𝑥𝑥(𝑋(𝐺𝑗))≤𝑐3âˆ‘âˆžğ‘˜=𝑘0𝐸𝐵𝑘(𝜏)𝜑(2−𝑘). Now, by [1, Lemma 4.1], 𝐸𝑁𝑘(𝑠)≤𝑐22𝑚, for a positive constant 𝑐2.
Thus, using (3.16), we have 𝐸𝜑−𝑝𝑥𝑥𝑋𝐺𝑗≤𝑐3âˆžî“ğ‘˜=𝑘0ℎ2−𝑘2𝑘→0a.s.as𝑘0→∞(3.17) since ∑((ℎ(2−𝑘))2/𝑘)<∞ if ∑(ℎ(2−𝑘)/𝑘)<∞ for ℎ(2−𝑘) sufficiently small.
It follows that 𝜑−𝑝𝑥𝑥𝑋(𝐺𝑗)=0 a.s., and from (3.8), 𝜑−𝑝𝑥𝑥𝑋(𝐺𝑗)=0 a.s. So 𝜑−𝑝𝑥𝑥⋃𝑋(âˆžğ‘—=1𝐺𝑗∑)≤𝜑−𝑝𝑥𝑥𝑋(𝐺𝑗)=0.
By (3.12), â‹ƒğºâˆªâˆžğ‘—=1𝐺𝑗=[0,𝜏], and therefore 𝜑−𝑝𝑋[0,𝜏]=0 if ∫0+(ℎ(𝑠)/𝑠ln(1/𝑠))𝑑𝑠<∞.
This completes the proof.

References

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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.


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