Small value probabilities via the branching tree heuristic

TL;DR

This paper provides intuitive proofs for small value probabilities in supercritical Galton-Watson processes and extends the approach to analyze intersection local times of Brownian motions, offering new insights into stochastic process behaviors.

Contribution

It introduces a simple, intuitive proof method for small value probabilities and applies this strategy to new contexts involving Brownian motion intersections.

Findings

Small value probabilities for Galton-Watson martingale limits derived.

Extension of proof strategy to Brownian motion intersection local times.

Resolved a problem on small value probabilities for Brownian intersections.

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 Schr\"oder and the B\"ottcher case. 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.