The singularity spectrum of Lévy processes in multifractal time

The singularity spectrum of the fish’s boundary

“…The boundary of the Fish intersects the horizontal axis at the points (−1/2, 0) and (1,0) and is symmetric with respect to this axis, since T commutes with the symmetry x −→ −x on T 1 . We shall then restrict our study to the upper half-Fish, whose boundary is denoted as a concave function F :…”

“…Since then, multifractal analysis has developed in many contexts, for instance in Probability Theory [16,1] (see [11,2,17] for other examples). In this article, we consider the example of a graph naturally appearing in an optimization problem in Ergodic Theory.…”

“…Given a real function f ∈ L ∞ loc on an open interval I and x 0 ∈ I, recall that f belongs to C α (x 0 ), for some α ≥ 0, if there exist a polynomial P of degree at most α and a constant C > 0 such that locally : (1) |f (x) − P (x − x 0 )| ≤ C|x − x 0 | α .…”

“…Let T 1 be the torus identified with R/Z and equipped with the transformation T (x) = 2x mod (1). Introduce the convex set M(T 1 , T ) of Borel probability measures on T 1 invariant by T , endowed with the weak * topology.…”

Random Wavelet Series Based on a Tree-Indexed Markov Chain

“…None of the multifractal stochastic processes studied up to now, such as the usual Lévy processes [22], the Lévy processes in multifractal time [4] or the random wavelet series with independent coefficients introduced by J.-M. Aubry and S. Jaffard [2,23], enjoys this property. The random wavelet series based on multifractal measures studied by J. Barral and S. Seuret [3] do not satisfy it either, even though their wavelet coefficients exhibit strong correlations.…”