Scaling exponents and multifractal dimensions for independent random cascades

Abstract

This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(#), and (ii) multifractal dimensions are considered for cascades with a generator vector (wι w c ) of the general type. These problems were previously studied for independent strongly bounded variables w l : 0 < a < w/ ^ c. Consequently, a broad class of models used in applications, including Kolmogorov's log-normal model in turbulence, log-stable "universal" cascades in atmospheric dynamics, has not been covered. Roughly speaking, problems (i), (ii) are here solved under the condition that the scaling exists; the τ-function is calculated for all arguments (previously this was done for positive q) and a new effect emerges: the τ-function can generally involve discontinuities in the first derivative as well as in the second.

Comparing multifractal formalisms: the neighboring boxes condition

“…In this section, we consider the random measures introduced by B. Mandelbrot in [24]. Up to now, their multifractal analysis has been mostly performed in the setting of the tree of c-adic intervals (see [22,21,14,30,2,3]). Nevertheless, Arbeiter and Patzschke [1] obtained a result in the same spirit as ours under strong assumptions without setting a general frame.…”

Renewal of singularity sets of random self-similar measures

“…As we said, multifractal analysis of µ [27,34,43,23,1,7,5] usually considers Hölder singularities sets of the form (1.1) and their Hausdorff dimension d µ (α), which is a measure of their size. The method used to compute d µ (α) is to find a random measure µ α (of the same nature as µ) such that µ α is concentrated on E µ α ∩ E µα τ * µ (α) .…”

“…Since E(W −h K ) < ∞, using the approach of [43] to study the behavior at ∞ of Laplace transforms satisfying an inequality like (4.6) (see also [4,38]…”

Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws

“…There is a huge mathematical literature devoted to the study of such a construction and we refer the reader to Refs. [20,21,22,23] for rigourous results about the existence, regularity and statistical properties of Mandelbrot cascades. In physics or other applied sciences, as recalled in the introduction, the previous construction (and many of its variants) is considered as the paradigm for multifractal objects and has been often used as a reference model in order to reproduce observed multiscaling.…”