Small value probabilities via the branching tree heuristic

Abstract

In the first part of this paper, we give easy and intuitive proofs for the small value probabilities of the martingale limit of a supercritical Galton–Watson process in both the Schröder and the Böttcher cases. These results are well known, but the most cited proofs rely on generating function arguments which are hard to transfer to other settings. In the second part, we show that the strategy underlying our proofs can be used in the quite different context of self-intersections of stochastic processes. Solving a problem posed by Wenbo Li, we find the small value probabilities for intersection local times of several Brownian motions, as well as for self-intersection local times of a single Brownian motion.