The singularity spectrum of Lévy processes in multifractal time

Abstract

Let X = ( X t ) 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 ( X F ( 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.