Stochastic 2 micro-local analysis

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

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