Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws).
Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an `additive cascade' of independent normal variables.
In this course, we propose an elementary and self-contained introduction to canonical Mandelbrot random cascades. The multiplicative construction is explained and the necessary and sufficient condition of non-degeneracy is proved. Then, we discuss the problem of the existence of moments and the link with nondegeneracy. We also calculate the almost sure dimension of the measures. Finally, we give an outline on multifractal analysis of Mandelbrot cascades. This course was delivered in september 2013 during a meeting of the "Multifractal Analysis GDR" (GDR no 3475 of the french CNRS).
We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $μ$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $μ(B_n(x))$ as $n \to \infty$.
We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $μ$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\T^nμ)_{n\ge 0}$ (this completes in a non trivial way our previous result in th…
We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than Hölder), all our results are sharp. Thi…
在这篇文章中，我们考虑了著名的Mandelbrot星方程的连续模拟与对数正态权重。曼德尔布罗特引入这个方程来描述乘法级联定律。我们证明了满足上述连续方程的测度的存在性和唯一性;这些测度属于J. P. Kahane在1985年开发的高斯乘性混沌理论（或该理论的可能扩展）的范围。作为一个产品，我们也得到了这些措施的协方差结构的明确表征。我们还证明了定性性质，如长程独立性或各向同性可以读出的方程。
