Antoine Ayache

The Generalized Multifractional Brownian Motion

It is well known that the fractional Brownian motion (FBM) is of great interest in modeling. However, its Hölder is the same all along its path and this restricts its field of application. Therefore, it would be useful to construct a Gaussian process extending the FBM and having a Hölder that is allowed to change. A partial answer to this problem is supplied by the multifractional Brownian motion (MBM); but the Hölder of the MBM must necessarily be continuous and this may be a drawback in some situations. In this paper we construct a Gaussian process generalizing the MBM and having a Hölder that can be a ‘very irregular’ function.

Multifractional processes with random exponent

Multifractional Processes with Random Exponent (MPRE) are obtained by replacing the Hurst parameter of Fractional Brownian Motion (FBM) with a stochastic process. This process need not be independent of the white noise generating the FBM. MPREs can be conveniently represented as random wavelet series. We will use this type of representation to study their Holder regularity and their self-similarity.

Generalized Multifractional Brownian Motion: Definition and Preliminary Results

The Multifractional Brownian Motion (MBM) is a generalization of the well known Fractional Brownian Motion. One of the main reasons that makes the MBM interesting for modelization, is that one can prescribe its regularity: given any Holder function H(t), with values in ]0, 1[, one can construct an MBM admitting at any t 0, a Holder exponent equal to H(t 0). However, the continuity of the function H(t) is sometimes undesirable, since it restricts the field of application. In this work we define a gaussian process, called the Generalized Multifractional Brownian Motion (GMBM) that extends the MBM. This process will also depend on a functional parameter H(t) that belongs to a set 풜, but 풜 will be much more larger than the space of Holder functions.

Wavelet construction of Generalized Multifractional processes.

We construct Generalized Multifractional Processes with Random Exponent (GMPREs). These processes, defined through a wavelet representation, are obtained by replacing the Hurst parameter of Fractional Brownian Motion by a sequence of continuous random processes. We show that these GMPREs can have the most general pointwise Hölder exponent function possible, namely, a random Hölder exponent which is a function of time and which can be expressed in the strong sense (almost surely for all t), as a lim inf of an arbitrary sequence of continuous processes with values in [0, 1].

Construction of non separable dyadic compactly supported orthonormal wavelet bases for L2(R2) of arbitrarily high regularity

By means of simple computations, we construct new classes of non separable QMFs. Some of these QMFs will lead to non separable dyadic compactly supported orthonormal wavelet bases for L2(R2) of arbitrarily high regularity.

Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times

By using a wavelet method we prove that the harmonisable-type N -parameter multifractional Brownian motion (mfBm) is a locally nondeterministic Gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.

Regularity and identification of Generalized Multifractional Gaussian Processes

In this article a class of multifractional processes is introduced, called Generalized Multifractional Gaussian Process (GMGP). For such multifractional models, the Hurst exponent of the celebrated Fractional Brownian Motion is replaced by a function, called the multifractional function, which may be irregular. The main aim of this paper is to show how to identify irregular multifractional functions in the setting of GMGP. Examples of discontinuous multifractional functions are also given.

Lp properties for Gaussian random series

Let c = (c n ) n ∈N* be an arbitrary sequence of l 2 (N*) and let F c (ω) be a random series of the type F c (ω) =Σn∈N* g n (ω)c n e n , where (g n ) n ∈N* is a sequence of independent N c (0,1) Gaussian random variables and (e n ) n ∈N* an orthonormal basis of L 2 (Y,Μ,μ) (the finite measure space (Y,M,μ) being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for F c (ω) to belong to L p (Y, M,μ), p ∈ [2, oo) for any c ∈ l 2 (N*) almost surely is that sup n∈ N* ∥e n ∥L p (Y,M,μ) < oo. One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrodinger equation posed on the open unit disc of R 2 .

Hausdorff dimension of the graph of the Fractional Brownian Sheet

Let {B(t)}t∈R d be the Fractional Brownian Sheet with multiindex α = (α1, . . . , αd), 0 < αi < 1. In [14], Kamont has shown that, with probability 1, the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube Q ⊂ R d is equal to d + 1 − min(α1, . . . , αd). In this paper, we prove that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.

Discretization error of wavelet coefficient for fractal like processes

Abstract Wavelet analysis has turned out to be a quite useful tool for estimating Hölder exponents as well as spectral densities of fractal processes, typically fractional Brownian motion (fBm) and related processes. In real life applications, the process is observed at discrete times and its (theoretical) wavelet coefficients have therefore to be replaced by their discretizations. This paper studies the discretization error of a wavelet coefficient of a stationary increments fractal Gaussian process belonging to a class which includes fBm and multiscale fBm. We obtain a harmonizable representation formula for the latter error and then optimal lower and upper bounds of its mean square. Our results remain valid under very mild assumption on the analyzing wavelet ψ: we need no vanishing moment condition (which is unusual in the wavelet setting), we only have to assume that the function ψ be compactly supported and Lipschitz continuous. By way of illustration, we present a short example of application to heartbeat time series and detection of time variation of the corresponding wavelet energy, with physiological interpretation.