Stochastic 2-microlocal analysis
  A lot is known about the H\"older 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 obtained: 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. 
Erick Herbin and Jacques L\'evy-V\'ehel