Branching on a Sierpinski graph

The descending motion of particles in a Sierpinski gasket subject to a branching process is examined. The splitting on escape nodes of falling particles makes the event of reaching the base of the gasket possible with positive probability. The r.v.’s Y(k)Y(k), representing the number of particles reaching level kk (that is the kk-th generation) is the main object of our analysis. The transition probabilities, the means and variances of Y(k)Y(k) are obtained explicitly with a number of recursive formulas concerning the probability generating functions EtY(k)EtY(k), k≥1k≥1. A section is also devoted to the analysis of extinction probabilities for the branching process developing in this specific fractal set.