Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics and Stochastic Analysis
VolumeΒ 2008, Article IDΒ 564601, 7 pages
http://dx.doi.org/10.1155/2008/564601
Research Article

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 𝑋𝑑={𝑋(𝑑),𝑑β‰₯0} be a one-dimensional symmetric Cauchy process. We prove that, for any measure function, πœ‘,πœ‘βˆ’π‘(𝑋[0,𝜏]) is zero or infinite, where πœ‘βˆ’π‘(𝐸) is the πœ‘-packing measure of 𝐸, thus solving a problem posed by Rezakhanlou and Taylor in 1988.

1. Introduction

Let 𝑋𝑑={𝑋(𝑑),𝑑β‰₯0} be a strictly stable Levy process taking values in 𝑅𝑛 (𝑛-dimensional Euclidean space) of index π›Όβˆˆ(0,2], that is, a Markov process with stationary independent increment whose characteristic function is given by 𝐸𝑒𝑖𝑒,𝑋𝑑=π‘’βˆ’π‘‘πœ“π›Όξ€·π‘’ξ€Έ.(1.1)

Here, 𝑒 and 𝑋𝑑 are points in 𝑅𝑛, (𝑒,π‘₯) is the ordinary inner product in 𝑅𝑛, and β€–π‘₯β€–2=(π‘₯,π‘₯). The Levy exponent πœ“π›Ό(𝑒) is of the form πœ“π›Όξ€·π‘’ξ€Έ=||𝑒||π›Όξ€œπ‘†π‘›π‘€π›Όξ€·ξ€Έπœ‡ξ€·ξ€Έπ‘’,𝑦𝑑𝑦,(1.2)where 𝑀𝛼=𝑒,𝑦1βˆ’π‘–sgn𝑒,𝑦tanπœ‹π›Ό2|||𝑒‖‖𝑒‖‖|||,𝑦𝛼𝑀if𝛼≠1,1ξ€·ξ€Έ=|||𝑒𝑒,𝑦‖‖𝑒‖‖|||+,𝑦2π‘–πœ‹ξ€·ξ€Έ|||ξ€·ξ€Έ|||.𝑒,𝑦log𝑒,𝑦(1.3)πœ‡(𝑑𝑦) is an arbitrary finite measure on the unit sphere 𝑆𝑛 in 𝑅𝑛, not supported on a diametrical plane. If in (1.2) πœ‡ is the uniform distribution on 𝑆𝑛, 𝑋𝑑 is called the isotropic stable Levy process with index 𝛼. In this case, πœ“π›Ό(𝑒)=πœ†|𝑒|𝛼 for some πœ†>0. When 𝛼=1, πœ‡ must also have the origin as its center of mass, that is, ξ€œπ‘†π‘›ξ€·ξ€Έπ‘¦πœ‡π‘‘π‘¦=0,(1.4)and the resulting process is the symmetric Cauchy process.

If βˆ«π‘†π‘›π‘¦πœ‡(𝑑𝑦)β‰ 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 𝑋𝑑 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 π‘…πœ generated by 𝑋𝑑 and defined by π‘…πœξ‚€ξ€Ίξ€»ξ‚==𝑋0,𝜏π‘₯βˆˆπ‘…π‘›ξ€·π‘‘ξ€Έξ€Ίξ€»ξ‚‡βˆΆπ‘₯=𝑋forsomeπ‘‘βˆˆ0,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 π›Όβˆˆ(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(𝑑) is a planar Brownian motion, 𝛼=𝑛=2, πœ‘βˆ’π‘[𝑋([0,𝑑])] is either zero or infinite for any measure function πœ‘. 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 πœ‘.

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 Ξ¦ be the class of functions: ξ€·ξ€Έβ†’ξ€·ξ€Έπœ‘βˆΆ0,𝛿0,∞(2.1) which are right continuous and monotone increasing with πœ‘(0+)=0 and for which there is a finite constant π‘˜>0 with πœ‘ξ€·ξ€Έ2π‘ πœ‘ξ€·π‘ ξ€Έπ›Ώβ‰€π‘˜for0<𝑠<2.(2.2)

The inequality (2.2) is a weak smoothness condition usually called a doubling property. A function πœ‘ in Ξ¦ is often called a measure function: ξ€·πΈξ€Έπœ‘βˆ’π‘ƒ=limsupπœ€β†’0ξ‚†ξ“π‘–πœ‘ξ‚€2π‘Ÿπ‘–ξ‚βˆΆπ΅ξ‚€π‘₯𝑖,π‘Ÿπ‘–ξ‚aredisjoint,π‘₯π‘–βˆˆπΈ,π‘Ÿπ‘–ξ‚‡<πœ€,(2.3) where 𝐡(π‘₯𝑖,π‘Ÿπ‘–) denotes the closure of the open ball 𝐡(π‘₯𝑖,π‘Ÿπ‘–) which is Β centered at π‘₯ and has radius π‘Ÿ.

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

πœ‘βˆ’π‘ƒ is a πœ‘-packing premeasure. The πœ‘-packing measure on 𝑅𝑛, denoted by πœ‘βˆ’π‘, is obtained by defining ξ€·πΈξ€Έξ‚†ξ“πœ‘βˆ’π‘=infπ‘›ξ€·πΈξ€Έξšπœ‘βˆ’π‘ƒβˆΆπΈβŠ†π‘›πΈπ‘›ξ‚‡.(2.4)It is proved in [2] that πœ‘βˆ’π‘(𝐸) is a metric outer measure, and hence every Borel set in 𝑅𝑛 is πœ‘βˆ’π‘ measurable.

We can see that for any πΈβŠ‚π‘…π‘›, ξ€·πΈξ€Έξ€·πΈξ€Έπœ‘βˆ’π‘β‰€πœ‘βˆ’π‘ƒ.(2.5)This gives a way to determine the upper bound of πœ‘βˆ’π‘(𝐸). Using the function πœ‘(𝑠)=𝑠𝛼, 𝛼>0 gives the fractal index dim𝑝𝐸=inf𝛼>0βˆΆπ‘ π›Όξ€·πΈξ€Έξ‚‡ξ‚†βˆ’π‘=0=sup𝛼>0βˆΆπ‘ π›Όξ€·πΈξ€Έξ‚‡βˆ’π‘=∞,(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 πœ‘βˆˆΞ¦, there exists a finite constant π‘˜>0 such that for any Borel measure πœ‡ on 𝑅𝑛 with 0<β€–πœ‡β€–=πœ‡(𝑅𝑛)<∞ and any Borel set πΈβŠ†π‘…π‘›, π‘˜βˆ’1πœ‡ξ€·πΈξ€Έinfπ‘₯βˆˆπΈξ‚†π·πœ‘βˆ’πœ‡ξ€·π‘₯ξ€Έξ‚‡βˆ’1ξ€·πΈξ€Έβ€–β€–πœ‡β€–β€–β‰€πœ‘βˆ’π‘β‰€π‘˜supπ‘₯βˆˆπΈξ‚†π·πœ‘βˆ’πœ‡ξ€·π‘₯ξ€Έξ‚‡βˆ’1,(2.7)where π·πœ‘βˆ’πœ‡ξ€·π‘₯ξ€Έ=liminfπ‘Ÿβ†“0πœ‡ξ‚€π΅ξ€·ξ€Έξ‚π‘₯,π‘Ÿπœ‘ξ€·ξ€Έ2π‘Ÿ(2.8) is the lower πœ‘-density of πœ‡ at π‘₯.

One then uses the sample path 𝑋𝑑 to define the random measure πœ‡ξ€·πΈξ€Έ=|||𝑑|||π‘‘βˆˆ0,πœβˆΆπ‘‹βˆˆπΈ(2.9) known as the occupation measure of the trajectory; |β‹…| denotes the Lebesgue measure.

This gives a Borel measure with πœ‡(𝑅)=‖𝑒‖=𝜏, and it is concentrated on 𝑋[0,𝜏] and spreads evenly on it.

If 𝑑π‘₯=𝑋0,0<𝑑0<𝜏,(2.10)then πœ‡ξ‚€π΅ξ€·ξ€Έξ‚=ξ€œπ‘₯,π‘Ÿπœ0𝐼𝐡π‘₯,π‘Ÿξ€Έξ‚€π‘‹ξ€·π‘‘ξ€Έξ‚ξ€·ξ€Έπ‘‘π‘‘=𝑇π‘₯,π‘Ÿ(2.11) is the sojourn time of 𝑋𝑑 in the ball 𝐡(π‘₯,π‘Ÿ) up to the time 𝜏. Define 𝜏=inf{𝑑>0∢|π‘₯(𝑑)|>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 π‘₯=0, one denotes 𝑇(π‘₯,π‘Ÿ) by 𝑇(π‘Ÿ).

In [8], we exhibited a measure function πœ‘ satisfying the following criteria.

Theorem 2.2. Suppose πœ‘=π‘Ÿβ„Ž(π‘Ÿ), where β„Ž(π‘Ÿ) is a monotone nondecreasing function and π‘‡ξ€·π‘Ÿξ€Έ=ξ€œπœ0𝐼𝐡0,π‘Ÿξ€Έξ‚€π‘‹ξ€·π‘‘ξ€Έξ‚π‘‘π‘‘;(2.12)then liminfπ‘Ÿβ†“0π‘‡ξ€·π‘Ÿξ€Έπœ‘ξ€·π‘Ÿξ€Έ=⎧βŽͺ⎨βŽͺβŽ©ξ€œ0if0+β„Žξ€·π‘ ξ€Έξ€·ξ€Έπ‘ ln1/𝑠=∞,∞otherwise,(2.13) where 𝑋𝑑 is a one-dimensional symmetric Cauchy process.

For any 𝑑0β‰₯0, 𝑋(𝑑+𝑑0)βˆ’π‘‹(𝑑0) is also a symmetric Cauchy process on the line since the finite-dimensional distribution of 𝑋(𝑑+𝑑0)βˆ’π‘‹(𝑑0) is independent of 𝑑0; see, for example, [1] for the strong Markov property of Cauchy processes.

The following corollary is then immediate.

Corollary 2.3. Let 𝑋𝑑, 𝑑β‰₯0, be a one-dimensional symmetric Cauchy process. Then, for any 𝑑0β‰₯0 with probability one, liminfπ‘Ÿβ†“0𝑇𝑋𝑑0,π‘Ÿπœ‘ξ€·π‘Ÿξ€Έ=⎧βŽͺ⎨βŽͺβŽ©ξ€œ0if0+β„Žξ€·π‘ ξ€Έξ€·ξ€Έπ‘ ln1/𝑠=∞,∞otherwise,(2.14) where πœ‘ is as defined in Theorem 2.2.

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

Lemma 2.4. Suppose πœ‘(π‘Ÿ)=π‘Ÿβ„Ž(π‘Ÿ), where β„Ž(π‘Ÿ) is a monotone increasing function. For 𝑇 defined in (2.12), then for any fixed constant 𝑐1 and π‘Žπ‘˜=πœŒβˆ’π‘˜, 𝜌>1, π‘ƒξ‚†π‘‡ξ‚€π‘Žπ‘˜+1<𝑐1πœ‘ξ‚€π‘Žπ‘˜ξ‚ξ‚‡β‰€π‘2β„Žξ‚€π‘Žπ‘˜ξ‚π‘˜.(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(𝑑)={𝑋(𝑑):𝑑β‰₯0} be a one-dimensional symmetric Cauchy process. If πœ‘(π‘Ÿ)=π‘Ÿβ„Ž(π‘Ÿ), where β„Ž is a nondecreasing function, then with probability one, 𝑋=⎧βŽͺ⎨βŽͺβŽ©ξ€œπœ‘βˆ’π‘0,πœξ‚ξ‚0if0+β„Žξ€·π‘ ξ€Έξ€·ξ€Έπ‘ ln1/𝑠<∞,∞otherwise,(3.1) where πœ‘βˆ’π‘(𝑋([0,𝜏])) is the πœ‘-packing measure of 𝑋([0,𝜏]).

Proof. In order to apply the density Lemma 2.1, we have to calculate liminfπ‘Ÿβ†“0πœ‡ξ‚€π΅ξ€·ξ€Έξ‚π‘₯,π‘Ÿπœ‘ξ€·ξ€Έ2π‘Ÿ.(3.2)But by Corollary 2.3, for each fixed 𝑑0∈(0,𝜏) with probability one, liminfπ‘Ÿβ†“0πœ‡ξ‚€π΅ξ‚€π‘‹ξ‚€π‘‘0,π‘Ÿξ‚ξ‚πœ‘ξ€·π‘Ÿξ€Έ=liminfπ‘Ÿβ†“0𝑇𝑋𝑑0,π‘Ÿπœ‘ξ€·π‘Ÿξ€Έξ€œ=0if0+β„Žξ€·π‘ ξ€Έπ‘ ln(1/𝑠)=∞.(3.3)Then a Fubini argument gives |||ξ‚†ξ€·ξ€Έπ‘‘βˆˆ0,𝜏∢liminfπ‘Ÿβ†“0πœ‡ξ‚€π΅ξ‚€π‘‹ξ€·π‘‘ξ€Έ,π‘Ÿξ‚ξ‚πœ‘ξ€·π‘Ÿξ€Έξ‚‡|||=0a.s.=𝜏<∞(3.4) so that if 𝐸={𝑋(𝑑0):𝑑0∈(0,𝜏)}, then πΈβŠ†π‘‹([0,𝜏]) and πœ‡(𝐸)=𝜏<∞ a.s. Using an application of the inequality of the density Lemma 2.1, we have ξ€·πΈξ€Έπœ‘βˆ’π‘=∞,(3.5)and thus πœ‘βˆ’π‘π‘‹([0,𝑑])=∞ with probability one if ∫0+(β„Ž(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 β€œbad-point” argument similar to that in [3].
For each point π‘₯βˆˆπ‘…, let π‘‰π‘˜(π‘₯) denote a semidyadic interval with length 2βˆ’π‘˜ whose complement is at distance 2βˆ’π‘˜βˆ’2 from a dyadic interval of length 2βˆ’π‘˜βˆ’2 which contains π‘₯.
Let Γ𝐸=ξ‚†π‘‰π‘˜ξ€·π‘₯ξ€Έξ‚‡βˆΆπ‘˜=1,2,…,π‘₯∈𝐸.(3.6)We use the intervals in Γ𝐸 to replace the balls 𝐡(π‘₯,π‘Ÿ) in (2.3) with length 2βˆ’π‘˜ replacing 2π‘Ÿ=diam𝐡(π‘₯,π‘Ÿ).
This gives a new premeasure πœ‘βˆ’π‘ƒπ‘₯π‘₯(𝐸) comparable to πœ‘βˆ’π‘ƒ as follows. There exist positive finite constants π‘˜1, π‘˜2 such that, for all Borel sets πΈβŠ‚π‘…, π‘˜1πœ‘βˆ’π‘ƒπ‘₯π‘₯ξ€·πΈξ€Έξ€·πΈξ€Έβ‰€πœ‘βˆ’π‘ƒβ‰€π‘˜2πœ‘βˆ’π‘ƒπ‘₯π‘₯𝐸,(3.7) where πœ‘βˆ’π‘π‘₯π‘₯𝐸=infπ‘–πœ‘βˆ’π‘ƒπ‘₯π‘₯ξ‚€πΈπ‘–ξ‚βˆΆπΈβŠ‚βˆͺ𝐸𝑖.(3.8)For ξ€œβˆž0+β„Žξ€·π‘ ξ€Έπ‘ ln(1/𝑠)<∞,(3.9)let 𝑑𝐺=0βˆˆξ€·ξ€Έπœ‡ξ‚€π΅ξ‚€π‘‹ξ‚€π‘‘0,𝜏∢liminf0,π‘Ÿξ‚ξ‚πœ‘ξ€·ξ€Έξ‚‡2π‘Ÿ=∞(3.10) be the set of β€œgood” points. A Fubini argument tells us that |𝐺|=𝜏<∞ a.s.; then using the density lemma in the other direction, we have ξ‚€π‘‹ξ€·πΊξ€Έξ‚πœ‘βˆ’π‘ƒ=0.(3.11)Let ξ€Ίξ€»0,𝜏⧡𝐺=βˆžξšπ‘–=1𝐺𝑗,(3.12)where 𝐺𝑗=ξ‚†ξ€·ξ€Έπœ‡ξ‚€π΅ξ‚€π‘‹ξ€·π‘‘ξ€Έπ‘‘βˆˆ0,𝜏∢liminf,π‘Ÿξ‚ξ‚πœ‘ξ€·ξ€Έξ‚‡2π‘Ÿβ‰€π‘—(3.13) is the set of β€œbad” points.
For π‘‘βˆˆπΊπ‘—, by monotonicity, we have for a positive constant 𝑐, πœ‡ξ‚€π΅ξ‚€π‘‹ξ€·π‘‘ξ€Έ,2βˆ’π‘˜ξ‚€2ξ‚ξ‚β‰€π‘π‘—πœ‘βˆ’π‘˜ξ‚,(3.14)for infinitely many π‘˜.
For fixed 𝑗, we can only get a contribution to πœ‘βˆ’π‘ƒπ‘₯π‘₯(𝑋(𝐺𝑗)) from semidyadic intervals of length 2βˆ’π‘˜ if the dyadic interval of length 2βˆ’π‘˜βˆ’2 is entered by 𝑋(𝑑) at time π‘‘β‰€πœ and the restarted process leaves the interval of length 2βˆ’π‘˜βˆ’2 in less than π‘—πœ‘(2βˆ’π‘˜).
Thus, if π‘†π‘˜ is a semidyadic interval of length 2βˆ’π‘˜, then π‘†π‘˜ is bad if 𝑋(𝑠) enters inside dyadic interval of length 2βˆ’π‘˜βˆ’2 but spends less than π‘—πœ‘(2βˆ’π‘˜) in π‘†π‘˜; otherwise it is β€œgood”. Any π‘‘βˆˆπΊπ‘— will be in infinitely many such bad π‘†π‘˜.
The probability that π‘†π‘˜ is bad given that it is entered is at most 𝑃𝑇2βˆ’π‘˜ξ‚ξ‚€2β‰€π‘π‘—πœ‘βˆ’π‘˜β‰€ξ‚€2ξ‚ξ‚‡π‘β„Žβˆ’π‘˜ξ‚π‘˜,(3.15) by Lemma 2.4.
Let π‘π‘˜(𝜏) be the number of intervals of length 2βˆ’π‘˜ that are entered by the time 𝜏, and let π΅π‘˜(𝜏) denote the number of those that are bad; then πΈπ΅π‘˜ξ€·πœξ€Έβ‰€πΈπ‘π‘˜ξ€·πœξ€Έβ„Žξ‚€2βˆ’π‘˜ξ‚π‘˜.(3.16)Leaving out the nonoverlapping requirement, we have, for a positive constant 𝑐3, πΈπœ‘βˆ’π‘π‘₯π‘₯(𝑋(𝐺𝑗))≀𝑐3βˆ‘βˆžπ‘˜=π‘˜0πΈπ΅π‘˜(𝜏)πœ‘(2βˆ’π‘˜). Now, by [1, Lemma 4.1], πΈπ‘π‘˜(𝑠)≀𝑐22π‘š, for a positive constant 𝑐2.
Thus, using (3.16), we have πΈπœ‘βˆ’π‘π‘₯π‘₯𝑋𝐺𝑗≀𝑐3βˆžξ“π‘˜=π‘˜0ξ‚€β„Žξ‚€2βˆ’π‘˜ξ‚ξ‚2π‘˜β†’0a.s.asπ‘˜0β†’βˆž(3.17) since βˆ‘((β„Ž(2βˆ’π‘˜))2/π‘˜)<∞ if βˆ‘(β„Ž(2βˆ’π‘˜)/π‘˜)<∞ for β„Ž(2βˆ’π‘˜) sufficiently small.
It follows that πœ‘βˆ’π‘π‘₯π‘₯𝑋(𝐺𝑗)=0 a.s., and from (3.8), πœ‘βˆ’π‘π‘₯π‘₯𝑋(𝐺𝑗)=0 a.s. So πœ‘βˆ’π‘π‘₯π‘₯⋃𝑋(βˆžπ‘—=1πΊπ‘—βˆ‘)β‰€πœ‘βˆ’π‘π‘₯π‘₯𝑋(𝐺𝑗)=0.
By (3.12), ⋃𝐺βˆͺβˆžπ‘—=1𝐺𝑗=[0,𝜏], and therefore πœ‘βˆ’π‘π‘‹[0,𝜏]=0 if ∫0+(β„Ž(𝑠)/𝑠ln(1/𝑠))𝑑𝑠<∞.
This completes the proof.

References

  1. Y. Xiao, β€œRandom fractals and Markov processes,” 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. 261–338, American Mathematical Society, Providence, RI, USA, 2004. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. S. J. Taylor and C. Tricot, β€œPacking measure, and its evaluation for a Brownian path,” Transactions of the American Mathematical Society, vol. 288, no. 2, pp. 679–699, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  3. S. J. Taylor, β€œThe use of packing measure in the analysis of random sets,” in Stochastic Processes and Their Applications, vol. 1203 of Lecture Notes in Mathematics, pp. 214–222, Springer, Berlin, Germany, 1986. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  4. F. Rezakhanlou and S. J. Taylor, β€œThe packing measure of the graph of a stable process,” Astérisque, no. 157-158, pp. 341–362, 1988. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. B. E. Fristedt and S. J. Taylor, β€œThe packing measure of a general subordinator,” Probability Theory and Related Fields, vol. 92, no. 4, pp. 493–510, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. J.-F. Le Gall and S. J. Taylor, β€œThe packing measure of planar Brownian motion,” in Seminar on Stochastic Processes, pp. 130–148, Birkhäuser, Boston, Mass, USA, 1986. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. R. Bañuelos and T. Kulczycki, β€œThe Cauchy process and the Steklov problem,” Journal of Functional Analysis, vol. 211, no. 2, pp. 355–423, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. A. C. Okoroafor, β€œSome local asymptotic laws for the Cauchy process on the line,” 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