Optimal survival strategy for branching Brownian motion in a Poissonian trap field

TL;DR

This paper investigates optimal survival strategies for branching Brownian motion in Poissonian trap fields, providing precise results on population size, range, and clearing configurations, and addressing open problems in the field.

Contribution

It establishes the first rigorous, precise survival strategies for branching Brownian motion in different Poissonian trap fields, including uniform and radially decaying cases.

Findings

Derived optimal survival strategies for different trap configurations.

Proved results on population size conditioned on survival.

Addressed open problems from previous literature.

Abstract

We study a branching Brownian motion $Z$ with a generic branching law, evolving in $\mathbb{R}^d$, where a field of Poissonian traps is present. Each trap is a ball with constant radius. We focus on two cases of Poissonian fields: a uniform field and a radially decaying field. Using classical results on the convergence of the speed of branching Brownian motion, we establish precise results on the population size of $Z$, given that it avoids the trap field, while staying alive up to time $t$. The results are stated so that each gives an 'optimal survival strategy' for $Z$. As corollaries of the results concerning the population size, we prove several other optimal survival strategies concerning the range of $Z$, and the size and position of clearings in $\mathbb{R}^d$. We also prove a result about the hitting time of a single trap by a branching system (Lemma 1), which may be useful in a completely generic setting too. Inter alia, we answer some open problems raised in [Mark. Proc. Rel. Fields, 9 (2003), 363 - 389].