Directed random walks on hierarchical trees with continuous branching: a renormalization group approach.

We investigate the directed random walk on hierarchic trees. Two cases are investigated: random variables on deterministic trees with a continuous branching, and random variables on the trees constructed through the random branching process. We derive renormalization group (partial differential) equations for the branching models with binomial, Poisson, and compound Poisson distributions of random variables on the links of a tree. These renormalization group equations are a new class of reaction-diffusion equations in one dimension.