Hausdorff dimension of the graph of an operator semistable Lévy process

Abstract

Let $X=\{X(t):t\ge 0\}$ be an operator semistable Lévy process in ${\mathbb{R}}^d$ with exponent $E$, where $E$ is an invertible linear operator on ${\mathbb{R}}^d$. For an arbitrary Borel set $B\subseteq {\mathbb{R}}_{+}$ we interpret the graph Gr${}_X(B)=\{(t,X(t)):tinB\}$ as a semi-selfsimilar process on ${\mathbb{R}}^{d+1}$, 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 [16] and [8].