RANDOM CASCADES ON WAVELET DYADIC TREES

We introduce a new class of random fractal functions using the orthogo nal wavelet transform. These functions are built recursively in the sp ace-scale half-plane of the orthogonal wavelet transform, ''cascading' ' from an arbitrary given large scale towards small scales. To each ra ndom fractal function corresponds a random cascading process (referred to as a W-cascade) on the dyadic tree of its orthogonal wavelet coeff icients. We discuss the convergence of these cascades and the regulari ty of the so-obtained random functions by studying the support of thei r singularity spectra. Then, we show that very different statistical q uantities such as correlation functions on the wavelet coefficients or the wavelet-based multifractal formalism partition functions can be u sed to characterize very precisely the underlying cascading process. W e illustrate all our results on various numerical examples. (C) 1998 A merican Institute of Physics. [S0022-2488(98)01008-1].