Abstract: We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super- Lévy process.

Key words: super-Lévy process, modulus of continuity, Hausdorff dimension, Lévy process, a-fast path, Brownian motion