Stochastic fixed-point equations

Abstract

The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.