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

Authors: A. C. Okoroafor
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

LetXt={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.


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## Body

## 1. Introduction

LetXt={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 (1.1)E[ei(u,Xt)]=e−tψα(u).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 (1.2)ψα(u)=|u|α∫Snwα(u,y)μ(dy),where (1.3)wα(u,y)=[1−i⁢sgn⁡(u,y)tan⁡(πα2)](|u‖u‖,y|)αifα≠1,w1(u,y)=|(u‖u‖,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, (1.4)∫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
pathsXt 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 setRτ generated by Xt and defined by (1.5)Rτ=X([0,τ])={x∈Rn: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 forα∈(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 forα=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].LetΦ be the class of functions: (2.1)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 (2.2)φ(2s)φ(s)≤kfor0<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: (2.3)φ−P(E)=lim⁡sup⁡ε→0{∑iφ(2ri):B¯(xi,ri)aredisjoint,xi∈E,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 φ−p, is obtained by defining (2.4)φ−p(E)=inf⁡{∑nφ−P(E):E⊆⋃nEn}.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 anyE⊂Rn, (2.5)φ−p(E)≤φ−P(E).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 (2.6)dim⁡p(E)=inf⁡{α>0:sα−p(E)=0}=sup⁡{α>0:sα−p(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 [2], which we will call Lemma 2.1.Lemma 2.1.
For  a givenφ∈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, (2.7)k−1μ(E)inf⁡x∈E{D−μφ(x)}−1≤φ−p(E)≤k‖μ‖sup⁡x∈E{D−μφ(x)}−1,where (2.8)D−μφ(x)=lim⁡inf⁡r↓0μ(B(x,r))φ(2r) is the lower φ-density of μ at x.One
then uses the sample pathXt to define the random measure (2.9)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 withμ(R)=∥u∥=τ,
and it is concentrated on X[0,τ] and spreads evenly on it.If(2.10)x=X(t0),0<t0<τ,then (2.11)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 [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.
Supposeφ=rh(r),
where h(r) is a monotone nondecreasing function and (2.12)T(r)=∫0τIB(0,r)(X(t))dt;then (2.13)lim⁡inf⁡r↓0T(r)φ(r)={0if∫0+h(s)sln⁡(1/s)=∞,∞otherwise, where Xt is a one-dimensional symmetric Cauchy process.For
anyt0≥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.
LetXt, t≥0,
be a one-dimensional symmetric Cauchy process. Then, for any t0≥0 with probability one, (2.14)lim⁡inf⁡r↓0T(X(t0),r)φ(r)={0if∫0+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 timeT(r),
taken from [8, Theorem 3.1].Lemma 2.4.
Supposeφ(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=ρ−k, ρ>1, (2.15)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.
LetX(t)={X(t):t≥0} be
a one-dimensional symmetric Cauchy process. If φ(r)=rh(r),
where h is a nondecreasing function, then with
probability one, (3.1)φ−p(X([0,τ]))={0if∫0+h(s)sln⁡(1/s)<∞,∞otherwise, where φ−p(X([0,τ])) is the φ-packing measure of X([0,τ]).Proof.
In  order to apply the density Lemma2.1, we have to calculate (3.2)lim⁡inf⁡r↓0μ(B(x,r))φ(2r).But by Corollary 2.3, for each fixed t0∈(0,τ) with probability one, (3.3)lim⁡inf⁡r↓0μ(B(X(t0),r))φ(r)=lim⁡inf⁡r↓0T(X(t0),r)φ(r)=0if∫0+h(s)sln⁡(1/s)=∞.Then a Fubini argument gives (3.4)|{t∈(0,τ):lim⁡infr↓0⁡μ(B(X(t),r))φ(r)=0a.s.}|=τ<∞ 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 (3.5)φ−p(E)=∞,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 Lemma2.1 in the other direction, as well as a “bad-point” argument
similar to that in [3].
For each pointx∈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(3.6)ΓE={Vk(x):k=1,2,…,x∈E}.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 premeasureφ−Pxx(E) comparable to φ−P as follows. There exist positive finite
constants k1, k2 such that, for all Borel sets E⊂R, (3.7)k1φ−Pxx(E)≤φ−P(E)≤k2φ−Pxx(E), where (3.8)φ−pxx(E)=inf⁡{∑iφ−Pxx(Ei):E⊂∪Ei}.For (3.9)∫0+∞h(s)sln⁡(1/s)<∞,let (3.10)G={t0∈(0,τ):lim⁡inf⁡μ(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 (3.11)φ−P(X(G))=0.Let (3.12)[0,τ]∖G=⋃i=1∞Gj,where (3.13)Gj={t∈(0,τ):lim⁡inf⁡μ(B(X(t),r))φ(2r)≤j} is the set of “bad” points.
Fort∈Gj,
by monotonicity, we have for a positive constant c, (3.14)μ(B(X(t),2−k))≤cjφ(2−k),for infinitely many k.
For fixedj, 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, ifSk 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 thatSk is bad given that it is entered is at most (3.15)P{T(2−k)≤cjφ(2−k)}≤ch(2−k)k, by Lemma 2.4.
LetNk(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 (3.16)EBk(τ)≤ENk(τ)h(2−k)k.Leaving out the nonoverlapping requirement,
we have, for a positive constant c3, Eφ−pxx(X(Gj))≤c3∑k=k0∞EBk(τ)φ(2−k).
Now, by [1, Lemma 4.1], ENk(s)≤c22m,
for a positive constant c2.
Thus, using (3.16), we have (3.17)Eφ−pxx(X(Gj))≤c3∑k=k0∞(h(2−k))2k⟶0a.s.ask0⟶∞ since ∑((h(2−k))2/k)<∞
if
∑(h(2−k)/k)<∞ for h(2−k) sufficiently small.
It follows thatφ−pxxX(Gj)=0  a.s.,
and from (3.8), φ−pxxX(Gj)=0 a.s.
So φ−pxxX(⋃j=1∞Gj)≤∑φ−pxxX(Gj)=0.
By (3.12), G∪⋃j=1∞Gj=[0,τ],
and therefore φ−pX[0,τ]=0
if
∫0+(h(s)/sln(1/s))ds<∞.
This completes the proof.



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*Source: 564601-2008-09-28.xml*