Narrow Results

Complex natural systems present characteristics of scalar invariance. This behavior has been experimentally verified and a large related bibliography has been reported. Multifractal Formalism is a way to evaluate this kind of behavior. In the past years, different numerical methods to estimate the multifractal spectrum have been proposed. These methods could be classified into those that originated from the wavelet analysis and others from numerical approximations like the Multifractal Detrended Fluctuation Analysis (MFDFA), proposed by Kantelhardt and Stanley. Recently, S. Jaffard and co-workers proposed the Wavelet Leaders (WL) method that exploits the potential of wavelet analysis and the efficiency of the Multiresolution Wavelet Schema.

In a previous work, we checked that both methods are equivalent for estimating fractal properties in a series from singular measures. Now, we apply MFDFA and WL to natural signals with self-similar structures, but unknown multifractal spectrum. We observe that some differences appear in their respective estimations, particularly when the signals are corrupted with fractional Gaussian noise.

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}_t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t).

We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents.

The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector $F=(f_1,\dots ,f_L)$ of $L$ functions, with $L\ge 2$, we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function $f_l$ has exactly a given value ${\alpha }_l$ of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of $F$. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.

We consider the (unique) mild solution $u(t,x)$ of a one-dimensional stochastic heat equation on $[0,T]\times ℝ$ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of $u(t,x)$, denoted by ${\partial }_{x}u(t,x)$, and its representation as a Feynman–Kac type closed form. The chaos expansion of ${\partial }_{x}u(t,x)$ makes it possible to find its (optimal) Hölder regularity especially in space.

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order $\alpha 𝜀\mathrm{\text{(}}1/2,1)$ are investigated in $L^p$ spaces for $p\ge 2$. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of $p$th moment. In particular, the $p$th moment Hölder regularities in time and $p$th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order $p\ge 2$ driven by a tempered fractional Brownian motion (TFBM) $B^{\sigma ,\lambda }(t)$ with $-1/2<\sigma <0$ and $\lambda >0$. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of $p$th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of $\lambda \to 0$, $\sigma \in (-1/2,0)$ or $\lambda >0$, $\sigma \to {\sigma }_0\in (-1/2,0)$. In particular, we obtain $p$th moment Hölder regularity in time and $p$th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise.

In an effort to characterize uniform and pointwise Hölder regularities, we obtain necessary decay rates and sufficient decay rates of continuous and discrete shearlet transform across scales. They are the same rates as those of the Hart Smith and continuous curvelet transforms. We then consider the situation where the regularity on a line in a non-parallel direction is significantly lower than the directional regularity along the line in a neighborhood of the line. Similar to that of the Hart Smith and continuous curvelet transforms, a set of necessary conditions for this direction of singularity is that the continuous shearlet transform decays half an order faster in directions "away" from the direction of the line and that the decay rate in directions "near" the line depends also on the horizontal distance from the line to the parallel line containing the center of the shearlet function. These similarities among the three transforms call for a general theory of transforms "with parabolic scaling."

In this paper, a wavelet multifractal analysis is developed which permits to characterize simultaneous singularities for a vector of functions. An associated multifractal formalism is introduced and checked for the case of functions involving self similar aspects.

In this paper, we introduce the concept of $d$-dilation (dual) pseudo-splines in $L^2(ℝ)$ with dilation $d\ge 2$, characterize the linear independence of the integer shifts, the subdivision schemes of polynomial reproduction and Hölder regularity for (dual) pseudo-splines with general dilation. We present a new method to determine the Hölder regularity of refinable functions with general dilation $d\ge 2$. Furthermore, we compare the regularities between pseudo-splines and dual ones.