Scale Invariant Infinitely Divisible Cascades

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic. It was recently shown and explained how and why multifractal analysis could be fruitfully placed in the general framework of infinitely divisible cascades. The aim of this contribution is to design processes, called Infinitely Divisible Cascading (IDC) 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. This communication focuses on the specific scale invariant case. To illustrate the powerfulness of the method, we mention that IDC processes can 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.