Fine regularity of Lévy processes and linear (multi)fractional stable motion

Paul Balança (1)

1 MAS - Mathématiques Appliquées aux Systèmes - EA 4037

In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multi-fractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

Mots clés: 2-microlocal analysis; Hölder regularity; multifractal spectrum; oscillating singularities; Lévy processes; linear fractional stable motion

Domaines: Probabilités [math.PR]

Paul Balança. Fine regularity of Lévy processes and linear (multi)fractional stable motion. Electronic Journal of Probability, 2014, ⟨10.1214/EJP.v19-3393⟩. ⟨hal-01291636⟩