Dimension results for sample paths of operator stable Lévy processes

Abstract

Let X ¼ fX ðtÞ; t 2 R þ g be an operator stable Le´vy process in R d with exponent B, where B is an invertible linear operator on R d : We determine the Hausdorff dimension and the packing dimension of the range X ð½0; 1Þ in terms of the real parts of the eigenvalues of B. r

Packing dimension profiles and Lévy processes

“…For related works see also [20,26]. Up to now, the case d < α has remained open, except when α = 2 and d = 1, that is, X is linear Brownian motion.…”

Hausdorff dimension of the graph of an operator semistable Lévy process

“…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]. …”

Fractal dimension results for continuous time random walks

“…The proof is similar to Proposition 4.1 in Meerschaert and Xiao (2005), using Lemma 2.7, and Propositions 2.5 and 2.6, hence we omit the details.…”

“…Theorem 3.2 in Meerschaert and Xiao (2005) shows that dim P Z ([0, α]) also equals the right hand side of (2.17). Then (2.16) follows using Theorem 2.2.…”

“…Now Theorem 3.2 in Meerschaert and Xiao (2005) implies that for any α > 0, $${\text{dim}}_{\mathrm{H}}Zfalse(false[0,afalse]false)={\text{dim}}_{\mathrm{P}}Zfalse(false[0,afalse]false)=\{\begin{array}{cc}leftleft2\beta & left\text{italicif}2\beta \le 1,\\ left\frac{1}{2}+\beta & left\text{italicif}2\beta >1,\end{array}\text{ a.s}.$$ …”