From N-parameter fractional Brownian motions to N-parameter multifractional Brownian motions

Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Holder regularity is allowed to vary along the paths. In this paper, two kind of multi-parameter extensions of mBm are studied: one is isotropic while the other is not. For each of these processes, a moving average representation, a harmonizable representation, and the covariance structure are given. The Holder regularity is then studied. In particular, the case of an irregular exponent function H is investigated. In this situation, the almost sure pointwise and local Holder exponents of the multi-parameter mBm are proved to be equal to the correspondent exponents of H. Eventually, a local asymptotic self-similarity property is proved. The limit process can be another process than fBm.Comment: 36 page

Weak convergence to the tangent process of the linear multifractional stable motion

We also show that one can have degenerate tangent processes Z(t), when the function H(t) is not sufficiently regular. The LMSM process is closely related to the Gaussian multifractional Brownian motion (MBM) process. We establish similar weak convergence results for the MBM

Invariance principle, multifractional Gaussian processes and long-range dependence

This paper is devoted to establish an invariance principle where the limit process is a multifractional Gaussian process with a multifractional function which takes its values in $(1/2,1)$. Some properties, such as regularity and local self-similarity of this process are studied. Moreover the limit process is compared to the multifractional Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP127 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.Comment: 35 page

From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields

Fine regularity of stochastic processes is usually measured in a local way by local H\"older exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,\rho)$ are bounded from above using the local H\"older exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the multiparameter fractional Brownian motion, the multifractional Brownian motion whose regularity function $H$ is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.Comment: 28 page

FracSim: An R Package to Simulate Multifractional Lévy Motions

In this article a procedure is proposed to simulate fractional fields, which are non Gaussian counterpart of the fractional Brownian motion. These fields, called real harmonizable (multi)fractional Lévy motions, allow fixing the Hölder exponent at each point. FracSim is an R package developed in R and C language. Parallel computers have been also used.

Linear Multifractional Stable Motion: fine path properties

Linear Multifractional Stable Motion (LMSM), denoted by $\{Y(t):t\in\R\}$, has been introduced by Stoev and Taqqu in 2004-2005, by substituting to the constant Hurst parameter of a classical Linear Fractional Stable Motion (LFSM), a deterministic function $H(\cdot)$ depending on the time variable $t$; we always suppose $H(\cdot)$ to be continuous and with values in (1/\al,1), also, in general we restrict its range to a compact interval. The main goal of our article is to make a comprehensive study of the local and asymptotic behavior of $\{Y(t):t\in\R\}$; to this end, one needs to derive fine path properties of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$, the field generating the latter process (i.e. one has $Y(t)=X(t,H(t))$ for all $t\in\R$). This leads us to introduce random wavelet series representations of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$ as well as of all its pathwise partial derivatives of any order with respect to $v$. Then our strategy consists in using wavelet methods. Among other things, we solve a conjecture of Stoev and Taqqu, concerning the existence for LMSM of a modification with almost surely continuous paths; moreover we provides some bounds for the local H\"older exponent of LMSM: namely, we obtain a quasi-optimal global modulus of continuity for it, and also an optimal local one. It is worth noticing that, even in the quite classical case of LFSM, the latter optimal local modulus of continuity provides a new result which was unknown so far