Infinitely divisible cascades

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic, to name but two. It was recently shown and explained how and why multifractal analysis could be interestingly and fruitfully placed in the more general framework of infinitely divisible cascades. The aim of this contribution is to design processes, called Infinitely Divisible Cascading noise, motion, and random walk. These processes possess at the same time stationary increments as well as multifractal and more general infinitely divisible scaling that can be prescribed a priori over a continuous range of scales. It also provides the reader with a thorough and detailed analysis of their statistical properties. A particular attention is paid to their scaling behaviours and a specific section is devoted to the scale invariant case. The construction is based on the general framework of Infinitely Divisible Cascades and general results are provided. The subclass of Compound Poisson Cascades is of particular interest from the points of view of both pedagogy and intuition as well as practical implementation and use. Specific results for Compound Poisson Cascades are therefore stated. To illustrate the powerfulness of the method, we conclude building processes that exactly mimic the scaling behaviors predicted by the celebrated She-Lévêque model of turbulence. Matlab routines implementing those processes are available from our WEB pages.