Let

Let

Here,

If

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

We are interested in the range of the
processes, that is, the random set

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

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 [

Further results on the asymmetric
Cauchy process and subordinators have been established by Rezakhanlou and
Taylor [

For the critical cases,

The main objective of this paper is
to show that for

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

Let

The
inequality (

A
sequence of closed balls satisfying the condition on the right side of (

We can see that for any

In
order to calculate the packing measure, we will use the following density
theorem of Taylor
and Tricot [

For a given

One
then uses the sample path

This
gives a Borel measure with

If

In [

Suppose

For
any

The following corollary is then immediate.

Let

One will also need an estimate for
the small ball probability of the sojourn time

Suppose

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.

In this section, we proceed to the main result.

Let

In order to apply the density Lemma

In order to prove the upper bound, we
use density Lemma

For each point

Let

This gives a new premeasure

For

For fixed

Thus, if

The probability that

Let

Thus, using (

It follows that

By (

This completes the proof.