Random cascades on wavelet dyadic trees

We introduce a new class of random fractal functions using the orthogonal wavelet transform. These functions are built recursively in the space-scale half-plane of the orthogonal wavelet transform, "cascading" from an arbitrary given large scale towards small scales. To each random fractal function corresponds a random cascading process (referred to as a W-cascade) on the dyadic tree of its orthogonal wavelet coefficients. We discuss the convergence of these cascades and the regularity of the so-obtained random functions by studying the support of their singularity spectra. Then, we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the wavelet-based multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process. We illustrate all our results on various numerical examples. (C) 1998 American Institute of Physics. [S0022-2488(98)01008-1].