Log-infinitely divisible multifractal processes

We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal "Multifractal Random Walk" processes (MRW) and the log-Poisson "product of cynlindrical pulses". Their construction involves some ``continuous stochastic multiplication'' from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.