Hausdorff dimension of the graph of an operator semistable Lévy process

Abstract

Abstract. Let X = {X(t) : t ≥ 0} be an operator semistable Lévy process in R d with exponent E, where E is an invertible linear operator on R d . For an arbitrary Borel set B ⊆ R + we interpret the graph Gr X (B) = {(t, X(t)) : t ∈ B} as a semi-selfsimilar process on R d+1 , whose distribution is not full, and calculate the Hausdorff dimension of Gr X (B) in terms of the real parts of the eigenvalues of the exponent E and the Hausdorff dimension of B. We use similar methods as applied in [12] and [6].

On the carrying dimension of occupation measures for self-affine random fields

“…Finally, in Section 4 we show that our results can be applied to large classes of self-affine random fields, namely to operator-self-similar stable random fields introduced by Li and Xiao [33], and to operator semistable Lévy processes. For these particular classes of self-affine random fields, our candidates derived by means of the singular value function in Section 3 are in fact the true values for the Hausdorff dimension of the graph and the range as recently shown in [45,46,28,47]. Furthermore, our results may be useful to derive Hausdorff dimension results for classes of random processes and fields, for which this still remains an open question, e.g.…”

“…. , m p , then recently Wedrich [47] (cf. also [27]) has shown that for any operator semistable Lévy process X almost surely…”

Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties

“…almost surely, and in case d = 1 by Theorem 3.2 in [34] for a strictly α-semistable Lévy process we have…”

“…almost surely. The derivation of (3.1)-(3.4) in [13,34] uses the standard method of showing that almost surely the right-hand side in (3.1)-(3.4) serves as an upper as well as a lower bound for the Hausdorff dimension on the corresponding lefthand side, following classical results for the range of one-dimensional stable Lévy processes in Blumenthal and Getoor [2,3,4] and Hendricks [9], and Lévy processes with independent stable components in Pruitt and Taylor [33,29,30]. The lower bound is shown by an application of Frostman's capacity theorem to prove that certain expected energy integrals are finite.…”

On exact Hausdorff measure functions of operator semistable Lévy processes

“…Lastly, we will also derive exact Hausdorff measure functions for type B which turned out to be more challenging than asserted in Remark 1 of [8]. The Hausdorff dimension for the range and the graph of operator semistable Lévy processes have recently been determined in [11] and [23], respectively; see also [10] for an alternative derivation based on an index formula presented in [14]. The special case of the limit process in subsequent cointossing games of the famous St. Petersburg paradox has been studied in [12] in detail.…”