Elliptic gaussian random processes

Abstract

We study the Gaussian random elds indexed by R d whose covariance is de ned in all generality as the parametrix of an elliptic pseudo-di erential operator with minimal regularity asumption on the symbol. We construct new wavelet bases adapted to these operators the decomposition of the eld on this corresponding basis yields its iterated logarithm law and its uniform modulus of continuity. We also characterize the local scalings of the eld in term of the properties of the principal symbol of the pseudodi erential operator. Similar results are obtained for the Multi-Fractional Brownian Motion.

Intermittency and multifractional Brownian character of geomagnetic time series

“…For this reason, in the second half of the 1990s the definition of fBm, characterized by a global value of H , was generalized about the case in which H is no longer independent of time, but a function of it. Consequently, in the generalization of fBm, i.e., mBm (Peltier and Lévy-Vehel, 1995;Benassi et al, 1997) Benassi et al (1997) with the same sequence of random numbers, so emphasize the differences in terms of fluctuations.…”

“…However, the pointwise regularity of a fBm is the same g its path, and this property is sometimes undesired, restricting the application field. For reason, in the second half of the 90s the definition of fBm, characterized by a global value of was generalized about the case in which H is no longer independent on time, but a function The fBm samples are generated from the algorithm described in Benassi et al (1997) with the same sequence of random numbers, so as to emphasize the differences in terms of fluctuations.…”

“…Starting from this definition, a direct scheme to generate a mBm has been implemented (Benassi et al, 1997;Muniandy and Lim, 2001). This scheme uses uncorrelated unit variance noise, i.e.,…”

“…2 displays a sample of mBm. These samples are generated from algorithms described in Benassi et al (1997) and Muniandy and Lim (2001), respectively. As it can be seen from the plots, the fBm and mBm fluctuations have a different character.…”

A Set-indexed Fractional Brownian Motion

“…B. Mandelbrot and J. W. Van Ness, and extended essentially into two directions. One is generally called multifractional Brownian motion, replacing the index parameter of self-similarity (called also the Hurst parameter) by a real measurable function (see [4], [18]). The other one are multiparameter fractional Brownian motions in which the set of the indices is a subset of the Euclidean space R N (see [14] and [22], [24] for trajectory properties).…”