The singularity spectrum of Lévy processes in multifractal time

Abstract

AbstractLet X=(Xt)t⩾0 be a Lévy process and μ a positive Borel measure on R+. Suppose that the integral of μ defines a continuous increasing multifractal time F:t⩾0↦μ([0,t]). Under suitable assumptions on μ, we compute the singularity spectrum of the sample paths of the process X in time μ defined as the process (XF(t))t⩾0.A fundamental example consists in taking a measure μ equal to an “independent random cascade” and (independently of μ) a suitable stable Lévy process X. Then the associated process X in time μ is naturally related to the so-called fixed points of the smoothing transformation in interacting particles systems.Our results rely on recent heterogeneous ubiquity theorems