JAMSA Journal of Applied Mathematics and Stochastic Analysis 1687-2177 1048-9533 Hindawi Publishing Corporation 564601 10.1155/2008/564601 564601 Research Article The Packing Measure of the Trajectory of a One-Dimensional Symmetric Cauchy Process Okoroafor A. C. Pourahmadi Mohsen Department of Mathematics Abia State University 440001 Uturu Nigeria abiastateuniversity.edu.ng 2008 25 08 2008 2008 03 08 2007 27 05 2008 12 08 2008 2008 Copyright © 2008 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.

Let Xt={X(t),t0} be a one-dimensional symmetric Cauchy process. We prove that, for any measure function, φ,φp(X[0,τ]) is zero or infinite, where φp(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),t0} 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)]=etψα(u).

Here, u and Xt are points in Rn, (u,x) is the ordinary inner product in Rn, and x2=(x,x). The Levy exponent ψα(u) is of the form ψα(u)=|u|αSnwα(u,y)μ(dy),where wα(u,y)=[1isgn(u,y)tan(πα2)](|uu,y|)αifα1,w1(u,y)=|(uu,y)|+2iπ(u,y)log|(u,y)|.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, Snyμ(dy)=0,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=X([0,τ])={xRn:x=X(t)forsomet[0,τ]}.

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, .

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, α<n, was solved by Taylor .

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

For the critical cases, α=n, the only known result is due to Le Gall and Taylor . They proved that if X(t) is a planar Brownian motion, α=n=2, φp[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 .

Let Φ be the class of functions: φ:(0,δ)(0,) which are right continuous and monotone increasing with φ(0+)=0 and for which there is a finite constant k>0 with φ(2s)φ(s)kfor0<s<δ2.

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: φP(E)=limsupε0{iφ(2ri):B¯(xi,ri)aredisjoint,xiE,ri<ε}, 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 φp, is obtained by defining φp(E)=inf{nφP(E):EnEn}.It is proved in  that φp(E) is a metric outer measure, and hence every Borel set in Rn is φp measurable.

We can see that for any ERn, φp(E)φP(E).This gives a way to determine the upper bound of φp(E). Using the function φ(s)=sα, α>0 gives the fractal index dimp(E)=inf{α>0:sαp(E)=0}=sup{α>0:sαp(E)=},called the packing dimension of E.

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

Lemma 2.1.

For a given φΦ, 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 ERn, k1μ(E)infxE{Dμφ(x)}1φp(E)kμsupxE{Dμφ(x)}1,where Dμφ(x)=liminfr0μ(B(x,r))φ(2r) is the lower φ-density of μ at x.

One then uses the sample path Xt to define the random measure μ(E)=|{t[0,τ]:X(t)E}| 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 x=X(t0),0<t0<τ,then μ(B(x,r))=0τIB(x,r)(X(t))dt=T(x,r) 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  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 , we exhibited a measure function φ satisfying the following criteria.

Theorem 2.2.

Suppose φ=rh(r), where h(r) is a monotone nondecreasing function and T(r)=0τIB(0,r)(X(t))dt;then liminfr0T(r)φ(r)={0if0+h(s)sln(1/s)=,otherwise, where Xt is a one-dimensional symmetric Cauchy process.

For any t00, 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,  for the strong Markov property of Cauchy processes.

The following corollary is then immediate.

Corollary 2.3.

Let Xt, t0, be a one-dimensional symmetric Cauchy process. Then, for any t00 with probability one, liminfr0T(X(t0),r)φ(r)={0if0+h(s)sln(1/s)=,otherwise, 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)=rh(r), where h(r) is a monotone increasing function. For T defined in (2.12), then for any fixed constant c1 and ak=ρk, ρ>1, P{T(ak+1)<c1φ(ak)}c2h(ak)k.

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):t0} be a one-dimensional symmetric Cauchy process. If φ(r)=rh(r), where h is a nondecreasing function, then with probability one, φp(X([0,τ]))={0if0+h(s)sln(1/s)<,otherwise, where φp(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 liminfr0μ(B(x,r))φ(2r).But by Corollary 2.3, for each fixed t0(0,τ) with probability one, liminfr0μ(B(X(t0),r))φ(r)=liminfr0T(X(t0),r)φ(r)=0if0+h(s)sln(1/s)=.Then a Fubini argument gives |{t(0,τ):liminfr0μ(B(X(t),r))φ(r)=0a.s.}|=τ< so that if E={X(t0):t0(0,τ)}, then EX([0,τ]) and μ(E)=τ< a.s. Using an application of the inequality of the density Lemma 2.1, we have φp(E)=,and thus φpX([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 .

For each point xR, let Vk(x) denote a semidyadic interval with length 2k whose complement is at distance 2k2 from a dyadic interval of length 2k2 which contains x.

Let ΓE={Vk(x):k=1,2,,xE}.We use the intervals in ΓE to replace the balls B(x,r) in (2.3) with length 2k replacing 2r=diamB(x,r).

This gives a new premeasure φPxx(E) comparable to φP as follows. There exist positive finite constants k1, k2 such that, for all Borel sets ER, k1φPxx(E)φP(E)k2φPxx(E), where φpxx(E)=inf{iφPxx(Ei):EEi}.For 0+h(s)sln(1/s)<,let G={t0(0,τ):liminfμ(B(X(t0),r))φ(2r)=} 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 φP(X(G))=0.Let [0,τ]G=i=1Gj,where Gj={t(0,τ):liminfμ(B(X(t),r))φ(2r)j} is the set of “bad” points.

For tGj, by monotonicity, we have for a positive constant c, μ(B(X(t),2k))cjφ(2k),for infinitely many k.

For fixed j, we can only get a contribution to φPxx(X(Gj)) from semidyadic intervals of length 2k if the dyadic interval of length 2k2 is entered by X(t) at time tτ and the restarted process leaves the interval of length 2k2 in less than jφ(2k).

Thus, if Sk is a semidyadic interval of length 2k, then Sk is bad if X(s) enters inside dyadic interval of length 2k2 but spends less than jφ(2k) in Sk; otherwise it is “good”. Any tGj will be in infinitely many such bad Sk.

The probability that Sk is bad given that it is entered is at most P{T(2k)cjφ(2k)}ch(2k)k, by Lemma 2.4.

Let Nk(τ) be the number of intervals of length 2k that are entered by the time τ, and let Bk(τ) denote the number of those that are bad; then EBk(τ)ENk(τ)h(2k)k.Leaving out the nonoverlapping requirement, we have, for a positive constant c3, Eφpxx(X(Gj))c3k=k0EBk(τ)φ(2k). Now, by [1, Lemma 4.1], ENk(s)c22m, for a positive constant c2.

Thus, using (3.16), we have Eφpxx(X(Gj))c3k=k0(h(2k))2k0a.s.ask0 since ((h(2k))2/k)< if (h(2k)/k)< for h(2k) sufficiently small.

It follows that φpxxX(Gj)=0 a.s., and from (3.8), φpxxX(Gj)=0 a.s. So φpxxX(j=1Gj)φpxxX(Gj)=0.

By (3.12), Gj=1Gj=[0,τ], and therefore φpX[0,τ]=0 if 0+(h(s)/sln(1/s))ds<.

This completes the proof.

Xiao Y. Lapidus M. L. van Frankenhuijsen M. Random fractals and Markov processes Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 2004 72 Providence, RI, USA American Mathematical Society 261 338 Proceedings of Symposia in Pure Mathematics MR2112126 ZBL1068.60092 Taylor S. J. Tricot C. Packing measure, and its evaluation for a Brownian path Transactions of the American Mathematical Society 1985 288 2 679 699 MR776398 10.2307/1999958 ZBL0537.28003 Taylor S. J. The use of packing measure in the analysis of random sets Stochastic Processes and Their Applications 1986 1203 Berlin, Germany Springer 214 222 Lecture Notes in Mathematics MR872112 10.1007/BFb0076883 ZBL0607.60033 Rezakhanlou F. Taylor S. J. The packing measure of the graph of a stable process Astérisque 1988 157-158 341 362 MR976226 ZBL0677.60082 Fristedt B. E. Taylor S. J. The packing measure of a general subordinator Probability Theory and Related Fields 1992 92 4 493 510 MR1169016 10.1007/BF01274265 ZBL0767.60009 Le Gall J.-F. Taylor S. J. The packing measure of planar Brownian motion Seminar on Stochastic Processes 1986 Boston, Mass, USA Birkhäuser 130 148 Bañuelos R. Kulczycki T. The Cauchy process and the Steklov problem Journal of Functional Analysis 2004 211 2 355 423 MR2056835 10.1016/j.jfa.2004.02.005 ZBL1055.60072 Okoroafor A. C. Some local asymptotic laws for the Cauchy process on the line Journal of Applied Mathematics and Stochastic Analysis 2007 2007 9 81934 MR2320642 10.1155/2007/81934