Uniform dimension results for a family of Markov processes

Abstract

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles in [59]. As applications, uniform Hausdorff and packing dimension results for self-similar Markov processes, certain classes of Lévy processes, stable jump diffusions and non-symmetric stable-type processes are obtained. general Markov processes related to Lévy processes. A large class of Markov processes are generated by pseudo-differential operators [8,20,21,54], their corresponding transition probabilities and heat kernel estimates have been studied in, for example, [38,40,41,46]. Also, we refer to [3] for the martingale problem of pure jump Markov processes, [9,10,11] for stable-like processes related to Dirichlet forms and their heat kernel estimates. Many natural questions regarding sample path and fractal properties arise for such Markov processes. Xiao [59] gives a comprehensive survey on fractal properties of Lévy or more general Markov processes before 2004. A lot of progress has been made since then. See, for example, [26,27,28,29,30,31] for various results on Lévy processes, [9,10,61] for Hausdorff dimension of the range or graph of stable-like processes, [37,36] for results on Hausdorff dimensions of the image, level and collision sets of a class of Feller process generated by a pseudo-differential operator. However, many interesting problems described [59] are still open.The main purpose of this paper is to provide a general method for establishing uniform Hausdorff and packing dimension results for more general Markov processes. In particular, we extend the methods of Hawkes [16] and Hawkes and Pruitt [18] for stable Lévy processes (see also Pruitt [48]) to a large family of Markov processes including more general Lévy processes, the stable-like processes or stable jump diffusions in [10,11,38,46]. The key technical tools are the covering principles in Section 2, which improve Lemmas 8.1 and 8.2 in Xiao [59]. We apply them to show the main result of this paper, Theorem 3.3, in Section 3. In Section 4, we apply Theorem 3.3 to Lévy processes and stabletype processes. We mention that Benjamini, Chen and Rohde [4] studied the normally reflected Brownian motion (RBM) in a class of non-smooth domains and proved their uniform Hausdorff dimension result by using the uniform Hölder continuity of RBM, the aforementioned Kaufman's theorem for Brownian motion and a subordination argument. Moreover, they indicated in Remark 3.10 in [4] that similar result still holds for the stable-like processes in [10] by using the covering principle in Xiao [59]. The scope of the present paper is a lot broader, and we expect that the main result of this paper will also be useful for studying fractal sets related to intersections and multiple points of Markov processes.In the rest of this paper, we assume that X = {X(t), t ∈ R + , P x } is a time-homogeneous Markov process with state space R d , defined on some probability space (Ω, F, P), and satisfies the...

Uniform Hausdorff dimension result for the inverse images of stable Lévy processes

“…Our method for proving the upper bound of dim H . 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.…”

“…Here we present an alternative argument. It follows from Hawkes and Pruitt [10] (see also [22]) that the following uniform dimension result holds:…”

“…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.…”

“…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…”

“…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.…”