Tree-indexed processes: A high level crossing analysis

Consider a branching diffusion process on R¹ starting at the origin. Take a high level u > 0 and count the number R(u, n) of branches reaching u by generation n. Let F_k,n(u) be the probability P(R(u, n) < k), k = 1, 2, . . .. We study the limit lim_n→∞ F _k,n (u) = F_k(u). More precisely, a natural equation for the probabilities F_k(u) is introduced and the structure of the set of solutions is analysed. We interpret F_k(u) as a potential ruin probability in the situation of a multiple choice of a decision taken at vertices of a 'logical tree'. It is shown that, unlike the standard risk theory, the above equation has a manifold of solutions. Also an analogue of Lundberg's bound for branching diffusion is derived.