A model for physicochemical and biological space-dependent branched chain processes is suggested based on the assumption that the number and the positions of the direct offspring of a particle are random variables characterized by a given point process. The generating functional of the random point process characterizing the number and the position of the total offspring from a given generation may be computed by repeated iteration of the generating functional attached to the offspring of a particle. A stochastic renormalization transformation is introduced by assuming that for each generation there is a constant probability that the growth of the population stops and that the actual state of the population is an average of the contributions of different generations. A detailed analysis is performed for supercritical self-similar branching processes for which the distance of a particle from its direct ancestor is a random variable selected from a symmetric jump probability density. In this case all Janossy and product densities of the population can be computed exactly. The fluctuations of particle concentrations for a given generation have an intermittent behavior. The asymptotic behavior of the total number N of particles is independent of the spatial distribution: the probabilty of N has a long tail of the inverse power law type modulated by a periodic function of lnN. In contrast, the spatial distribution of particles depends both on the jump probability density and on the total number of particles. If the moments of the jump probability density are finite then the positions ${\mathbf{r}}_1$,...,${\mathbf{r}}_{\mathit{N}}$ of N particles are Gaussian random variables with a correlation matrix obeying a logarithmic scaling law 〈${\mathbf{r}}_{\mathit{i}}$⋅${\mathbf{r}}_{\mathit{j}}$〉∼${\mathrm{\delta }}_{\mathit{i}\mathit{j}}$lnN, as N→∞. If the jump probability density has a long tail then the probability density of ${\mathbf{r}}_1$,...,${\mathbf{r}}_{\mathit{N}}$ for N→∞ is given by a separable Lévy law with a dimensional parameter given by a second scaling law b∼(lnN${)}^{1/\mathrm{\varepsilon }}$ as N→∞, where ɛ is the fractal exponent of the Lévy law.

• Received 12 September 1994

DOI:https://doi.org/10.1103/PhysRevE.51.3104