Analytic Multivariate Generating Function for Random Multiplicative Cascade Processes

Abstract

Institut fu¨r Hochenergiephysik, Osterreichische Akademie der Wissenschaften, Nikolsdorf¨ ergasse 18, A–1050 Vienna, Austria(12 December 1997 chao-dyn/9804024)We have found an analytic expression for the multivariategenerating function governing all n-point statistics of randommultiplicative cascade processes. The variable appropriate forthis generating function is the logarithm of the energy density,lnǫ, rather than ǫ itself. All cumulant statistics become sumsover derivatives of “branching generating functions” which areLaplace transforms of the splitting functions and completelydetermine the cascade process. We show that the branch-ing generating function is a generalization of the multifractalmass exponents. Two simple models from fully developedturbulence illustrate the new formalism.47.27.Eq, 02.50.Sk, 05.40.+j