The author studies the solution of the functional equation \(g=(f\circ g)^{*c}\) where * and \(\circ\) denote the additive and the multiplicative convolution, respectively, and where f and g are two probability measures on \(R^+\). This equation has been introduced by Mandelbrot in order to refine a model of turbulence suggested by Kolmogorov and Yaglom. By using a theorem of H. Kesten [Acta Math. 131, 207-248 (1973; Zbl 0291.60029)] the author shows the existence of solutions similar to stable laws on the one hand, and solutions analogous to the Gaussian law on the other hand. An iterative procedure is utilized to characterize these limiting laws.