Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties

Abstract

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on ${\mathbb{R}}^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy exponent, and certain index formulae.

References in corpus (4)

• Discrete scale invariance and complex dimensions
• Levy processes: Capacity and Hausdorff dimension
• Dynamical percolation on general trees
• Hausdorff dimension of operator semistable Lévy processes

Cited by in corpus (1)

• On self-similar Bernstein functions and corresponding generalized fractional derivatives