On the Double Points of Operator Stable Lévy Processes

Abstract

20 pages.International audienceWe determine the Hausdorff dimension of the set of double points for a symmetric operator stable L\'evy process in terms of the eigenvalues of its stability exponent

Multiple Points of Operator Semistable Lévy Processes

“…Analogous asymptotics for operator semistable Lévy processes were obtained in [14, (2.1),(2.5),(2.6),(2.7)]. Therefore, the proof is the same as in [23]. The last statement is justified at the beginning of this section.…”

“…We also give a necessary and sufficient condition for the existence of k-multiple points. Our results extend to all k ≥ 2 the recent work [23], where the set of double points (k = 2) was studied in the symmetric operator stable case.…”

“…The problem was originally studied for Brownian motion by Dvoretzky, Erdös, Kakutani and Taylor [4,5,6]. Their results were later extended to more general Lévy processes using various techniques, see [7,8,9,11,12,15,18,19,21,23,26,28,29,30] and the references therein. Let X = {X(t), t ∈ R + } be a stochastic process with values in R d , d ≥ 1, and let k ≥ 2 be an integer.…”

“…We denote by M k the set of kmultiple points of X. The objective of this paper is to study the set M k for symmetric operator semistable Lévy processes and to extend to all k ≥ 2 the recent results of [23], where the set of double points was investigated when X is symmetric operator stable. Recall from [25,27] that a Lévy process X is called operator semistable if the distribution ν of X(1) is full (i.e.…”

“…Our goal in this section is to prove Theorem 1 by extending the method of [23] in order to deal with the set M k for k ≥ 3. To this end we fix 2 ≥ α 1 ≥ α 2 > 0 and let X = {X(t), t ∈ R + } be a symmetric operator semistable Lévy process in R 2 with stability exponent B whose eigenvalues have real parts α −1 1 , α −1 2 , as explained in the Introduction.…”