Stochastic 2-microlocal analysis
The aim of the paper is to provide new tools for the fine characterization of the regularity of stochastic processes, and in particular, Gaussian processes, using the theory of stochastic 2-microlocal analysis. Basically, 2-microlocal analysis allows describing how the pointwise regularity of a function evolves under the action of (pseudo-)differential operators and in the theory of stochastic processes it can be used to study the pointwise regularity. It is proved that an upper bound on moments on the increments of a process around the point provides an almost sure lower bound for the 2-microlocal frontier at this point. A related uniform result is also established. For Gaussian processes, more precise result is obtained: the behavior of the incremental covariance allows to get the almost sure value of the 2-microlocal frontier at any given point. As an application, new and refined regularity properties are obtained for fractional and multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.
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