Stochastic 2-microlocal analysis
type: Item
instance of: scholarly article
title: Stochastic 2-microlocal analysis
zbMATH Open document ID: 1175.60032
author: Erick Herbin
author: Jacques Lévy-Véhel
published in: Stochastic Processes and their Applications
publication date: 2009-07-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0504551
review text: 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.
Mathematics Subject Classification ID: 60G05
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 5578976
zbMATH Keywords: 2-microlocal analysis
zbMATH Keywords: (multi)fractional Brownian motion
zbMATH Keywords: Gaussian processes
zbMATH Keywords: Hölder regularity
zbMATH Keywords: multiparameter processes
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
OpenAlex ID: W2083411800
arXiv ID: math/0504551
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DOI: 10.1016/J.SPA.2008.11.005
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description: scientific article