Anomalous features arising from random multifractals

Under the formalism of annealed averaging of the partition function, two ty pes of random multifractal measures with their probability of multipliers s atisfying power law distribution and triangular distribution are investigat ed mathematically. In these two illustrations, branching emerges in the cur ve of generalized dimensions, and more abnormally, negative values of gener alized dimensions arise. Therefore, we classify the random multifractal mea sures into three classes based on the properties of generalized dimensions. Other equivalent classifications are also presented by investigating the l ocation of the zero-point of tau (q) or the relative position either betwee n the f(alpha) curve and the diagonal f (alpha) = alpha or between the f(q) curve and the alpha (q) curve. We consequently propose phase diagrams to c haracterize the classification procedure and distinguish the scaling proper ties between different classes. The branching phenomenon emerging is due to the extreme value condition and the convergency of the generalized dimensi ons at point q = 1. We conjecture that the branching condition exists and t hat the classification is universal for any random multifractals. Moreover, the asymptotic behaviors of the scaling properties are studied. We apply t he cascade processes studied in this paper to characterizing two stochastic processes, i.e. the energy dissipation field in fully developed turbulence and the droplet breakup in atomization. The agreement between the proposed model and experiments are remarkable.