Stochastic 2 micro-local analysis

Erick Herbin (1, 2) , Jacques Lévy Véhel (1, 2)

1 Regularity - Probabilistic modelling of irregularity and application to uncertainties management
2 MAS - Mathématiques Appliquées aux Systèmes - EA 4037

A lot is known about the Hölder regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-)differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be otained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.

Mots clés: Multi-parameter processes; 2-microlocal analysis; (multi)fractional Brownian motion; Gaussian processes; Hölder regularity; Multi-parameter processes.

Domaines: Probabilités [math.PR]

Erick Herbin, Jacques Lévy Véhel. Stochastic 2 micro-local analysis. Stochastic Processes and their Applications, 2009, 119 (7), pp.2277-2311. ⟨10.1016/j.spa.2008.11.005⟩. ⟨inria-00538965⟩