Uniform Hausdorff dimension result for the inverse images of stable L\'evy processes

We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha$-stable L\'evy processes with $1<\alpha\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from that of Kaufman and depends on covering principles for Markov processes.


Introduction
Let X = {X(t), t ≥ 0, P x } be a real-valued strictly α-stable Lévy process with α ∈ (0, 2]. Its characteristic exponent is given by, for ξ ∈ R, with some constants σ > 0 and β ∈ [−1, 1] which are respectively the scale parameter and the skewness parameter. Throughout log = log e denotes the natural logarithm. Notice that, in the case of α = 1, X is a symmetric Cauchy process. When α = 2, X is a (scaled) Brownian motion. For 0 < α < 2, X shares the properties of self-similarity, independence and stationarity of increments, with Brownian motion, but it has heavy-tailed distributions and its sample functions are discontinuous. As such, stable Lévy processes form an important class of Markov processes. Many authors have studied the asymptotic and sample path properties of Lévy processes. We refer to the monographs [2] and [21] for systematic accounts on Lévy processes, and to [24,26] for information on their fractal properties.
This note is concerned with a uniform Hausdorff dimension result, Theorem 1.1, for the inverse images of real-valued strictly α-stable Lévy processes and is motivated by the following results of Hawkes [8] and Kaufman [11].
Hawkes [8] considered the Hausdorff dimension of the inverse image X −1 (F ) = {t ≥ 0 : X(t) ∈ F } and proved that if 1 ≤ α ≤ 2 and F ⊆ R is a fixed Borel set, then for every x ∈ R, Note that the null event on which (1.1) does not hold depends on F . It is natural to ask if the following uniform Hausdorff dimension result holds: For every x ∈ R, Such a result, when it is valid, is more useful than (1.1) because, outside of a single null event, the dimension formula holds not only for all deterministic Borel sets F ⊂ R but also for random sets F that depend on the sample path of X.
We claim that, in the case 0 < α < 1, there is no uniform result like (1.2). This is because The referee has asked us the following question that complements the aforementioned claim: 1 For every x ∈ R, does Here C is the family of all deterministic Borel sets F ⊂ R with dim H F ≥ 1−α. To answer this question, we first recall Theorem 2 of Hawkes [8] : If 0 < α < 1 and F ⊂ R is deterministic and satisfies dim H F ≥ 1 − α, then [8, p.93] for the notation), then The answer to the referee's question is "yes" because we can choose a Borel set F ∈ C such that F \F * is polar for X (cf. [8, p.96]), then it follows from Hawkes' result (iii) that for any x ∈ R the probability in (1.3) is not more than Motivated by the referee's question and Hawkes' result (1.5), one may further ask to characterize the family G of deterministic Borel sets F such that for some x ∈ R (depending on G), This question seems to be rather nontrivial. We can imagine that (1.7) may hold for certain family of self-similar sets on R, but this goes beyond the scope of the present paper. Our objective of this paper is to study the uniform dimension problem (1.2) for 1 ≤ α ≤ 2. The validity of (1.2) in the case α = 2 (X is a Brownian motion) is due to Kaufman [11]. His proof relies on the uniform modulus of continuity of Brownian motion as well as the Hölder continuity of the Brownian local time in the time variable. For 1 ≤ α < 2, the sample paths of an α-stable Lévy process are discontinuous, hence Kaufman's method is not applicable.
In the special case of F = {z}, it follows from Barlow et al [1, (8.7)] that if 1 < α ≤ 2 then This gives a uniform Hausdorff dimension result for the level sets of X. However, for 1 ≤ α < 2, it had been an open problem to prove (1.2) for all Borel sets F ⊆ R; see [26,Sec. 8.2] for a discussion.
In this note, we verify (1.2) by proving the following theorem.
As mentioned above, the case of α = 2 has already been proved by Kaufman [11] whose proof relies on special properties of Brownian motion. Our proof of Theorem 1.1 provides an alternative proof of his theorem.
The proof is split naturally into the upper bound part and lower bound part. To show the upper bound, we design a new covering principle (see Lemma 2.2 below) for the inverse images of recurrent processes (thus it is applicable to α = 1). This covering lemma constitutes the key technical contribution of the present paper, and we expect it to be useful for other discontinuous Markov processes. Note that Lemma 2.2 in this paper is different from the covering lemma of [22,Lemma 2.2], which is only applicable to transient Markov processes (see Remark 2.3 in Section 2 of this paper). To prove the lower bound in (1.2), we make use of the uniform modulus of continuity (in time) of the maximum local time of X due to Perkins [18], together with a covering principle for the range of X in [10,26,22]. Since X has no local time when α = 1, the proof of the lower bound in Theorem 1.1 is valid only for 1 < α ≤ 2. We think that (1.2) holds for α = 1 as well, but have not been able to give a complete proof.

Proof of the upper bound
In this section we assume that 1 ≤ α ≤ 2. We will show that For any Borel set B, we denote by T B the first hitting time of B by the process X. We state an asymptotic result due to Port [ (1). If 1 < α ≤ 2, then for any bounded interval B and any x ∈ R, where L B (x) is bounded from above on compact sets and is positive for x ∈ B, the closure of the set B. Here, f (t) ∼ g(t) means lim t→∞ f (t)/g(t) = 1.
(2). If α = 1, then for any bounded interval B and any x ∈ R, where L B (x) is bounded from above on compact sets and is positive for x ∈ B.
The main tool to obtain our upper bound is the following covering lemma. Before stating this lemma, we introduce some notation. Let U n be any partition of R with intervals of length 2 −n and D n be any partition of R + with intervals of length 2 −nα . The choices of partitions have no effect on the result. Therefore, Note by spatial homogeneity and scaling, we have that Due to the right continuity of the sample paths, we have X(τ k−1 ) ∈ U as τ k−1 < T . By the strong Markov property, we obtain By induction, we obtain Next we show that there exists a constant c T such that p n ≤ 1 − c T 2 −nα(1− 1 α ) . By the independence of increments and the fact that X(1) is supported on R ([23, Thm. 1]), as desired. For n, K ≥ 1, define the event A δ n by T ] cannot be covered by 2 nδ intervals of length 2 −nα .
Here U ∈ U n ∩ [−K, K] means that U ∈ U n and U ⊂ [−K, K]. We have for δ > α − 1, Since any interval of length 2 −nα is covered by two intervals from D n , the conclusion for all U ⊂ [−K, K] follows from the Borel-Cantelli Lemma. Letting K → ∞ completes the proof.
(2) Now consider α = 1. The proof of this case is basically the same as that of Part (1), except that 1 − p n ≥ c T /n by Lemma 2.1.(2), and We omit the details.

Remark 2.3. As is said in the Introduction, the covering principle in [22, Lemma 2.2] is not applicable here. Intuitively, a recurrent process visits a fixed interval infinitely often, hence we could not expect that the inverse images could be covered by finite number of intervals.
Mathematically, the condition in [22] is for some δ, p > 0 and ∞ n=1 r p n < ∞, which is not satisfied for recurrent Markov processes.
Let us prove the upper bound (2.1).
Fix a T > 0 for now. By Lemma 2.2, each X −1 (U i ) ∩ [0, T ] can be covered by 2 · 2 n i δ intervals {I i,k } (of length 2 −n i α ) in D n i , thus we see that and T ↑ ∞ yields the desired upper bound. Now we consider the case of α = 1. One could repeat the argument above and use Lemma 2.2 to get the desired conclusion. Here we present an alternative argument. It follows from Hawkes and Pruitt [10] (see also [22]) that the following uniform dimension result holds:

Proof of the lower bound
We assume that 1 < α ≤ 2. It follows from Kesten [12] and Hawkes [9] that X hits points and has local times {L x t , t ≥ 0, x ∈ R}. The local times characterize the sojourn properties of X via the occupation density formula: For all t ≥ 0 and all Borel measurable function Moreover, there is a version of the local times, still denoted by {L x t , t ≥ 0, x ∈ R}, which is jointly continuous in (t, x); see e.g., [2,16].
We use the Hölder continuity of the local times of X to prove the uniform lower bound for the inverse image sets. This approach has been previously used by Kaufman [11], which was extended by Monrad and Pitt [17] in their study of inverse images of recurrent Gaussian fields. In both articles, the uniform modulus of continuity of the sample paths were used. Since the sample paths of the α-stable Lévy process X are discontinuous, we will apply a covering principle in [26,22] for the range of X. Denote C n any partition of R + of intervals of length 2 −n . We recall here the covering principle, tailored to our situation.
There exists a finite positive integer K, such that P x -a.s., for all n large enough, X(I) can be covered by K intervals of diameter 2 · 2 −nγ , for all I ∈ C n .
Let L * ([s, t]) = sup x∈R (L x t − L x s ) be the maximum local time of X on [s, t]. We recall now the following result due to Perkins [18] on the uniform modulus of continuity (in time) of the maximum local time of a strictly α-stable Lévy process X with index α ∈ (1, 2].   [14,15,16] for more sample path properties (in the space variable) of the local times of symmetric Markov processes.
We are ready to give the proof of the lower bound in Theorem 1.1.
Proof of Theorem 1.1: lower bound. It suffices to consider compact set F . For any compact F ⊂ R and ε > 0, by Frostman's lemma (cf. [6]) there exists a probability measure µ supported on F such that µ(B) ≤ |diam(B)| dim H F −ε for any interval B ⊂ R with |B| ≤ 1. Define the random measure λ by It is clear that λ(dt) is supported on X −1 (F ) ⊂ R + , λ(R + ) > 0, and Let n be sufficiently large, we have by Lemma 3.2 that uniformly for a ∈ [0, 1 − 2 −n ]. On the other hand, by Lemma 3.1, there exist a sequence of intervals {I i } 1≤i≤K of length 2 −nγ with γ < 1/α such that the closure of X([a, a + 2 −n ]) is covered by the union of I i , therefore, We thus obtain It follows that λ(B) ≤ diam(B) 1− 1 α +γ dim H F −2ε for all Borel sets B with sufficiently small diameter. This and Frostman's lemma imply that Letting γ ↑ 1 α , then ε ↓ 0 yields the desired lower bound for dim H X −1 (F ). This finishes the proof of Theorem 1.1.

Concluding remarks
This note raises several interesting questions for further investigation. In the following, we list three of them and discuss briefly the main difficulties. Solutions of these questions will require developing new techniques for Lévy processes.
(i). As having mentioned in the Introduction, we think that Theorem 1.1 holds for α = 1.
However, without a local time, it is not clear to us how to construct a random Borel measure supported on X −1 (F ) such that Frostman's lemma is applicable. (ii). In [20,Thm. 22.1], the asymptotic result for the hitting times was obtained for recurrent Lévy processes with regularly varying λ-potential densities, see also the recent development by Grzywny  . We believe that a similar result also holds for a large class of more general Markov processes including stable jump diffusions, stable like processes and Lévy-type processes as considered in [22]. However, proving such a result would require establishing first the asymptotic results for the hitting times and local times of these Markov processes. This is pretty challenging and goes well beyond the scope of the present paper. We will try to tackle this in a subsequent paper. (iii). It is natural to expect that the packing dimension analogue of Theorem 1.1 also holds.
Namely, if X is a real-valued strictly α-stable Lévy process with 1 ≤ α ≤ 2, then for any x ∈ R one has P x dim P X −1 (F ) = 1 − 1 α + dim P F α for all Borel sets F ⊆ R = 1. (4.1) Here dim P denotes packing dimension; see Falconer [6,Chapter 3] for its definition and properties, and [24,26] for examples of its applications in studying sample path properties of Markov processes. By using the connection between packing dimension and the upper box-counting (Minkowski) dimension (cf. [6]), one can see that the proof of the upper bound of Theorem 1.1 also implies that P x -a.s., In order to prove the reverse inequality, one may apply the lower density theorem for packing measure in [25,Theorem 5.4] and prove that for any γ < 1/α and ε > 0, where λ is the random measure defined in (3.2) and c 2 is a finite constant. We are not able to prove this because (unlike the Hausdorff dimension case) the terms µ(I i ) in (3.3) can not be controlled for all i by the same n.