[Paper Review] Branching Brownian motion with "mild" Poissonian obstacles

János Engländer|ArXiv.org|Aug 29, 2005

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper studies branching Brownian motion with 'mild' Poissonian obstacles—regions where particle reproduction is blocked but particles survive. It establishes that the quenched local growth rate equals the branching rate in the free region, and derives precise asymptotics for both quenched (d ≤ 2) and annealed (any d) global growth rates, identifying subexponential correction terms.

ABSTRACT

We study a spatial branching model, where the underlying motion is Brownian motion and the branching is affected by a random collection of reproduction blocking sets called "mild" obstacles. We show that the quenched local growth rate is given by the branching rate in the `free' region . When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the local growth that is independent of the Poissonian intensity. Finally, and most importantly, we obtain the asymptotics (in probability) of the quenched (when $d\le 2$) and the annealed (arbitrary d) global growth rates, and identify subexponential correction terms.

Motivation & Objective

• To analyze the long-term behavior of branching Brownian motion in a random environment defined by Poissonian obstacles that suppress reproduction but not particle survival.
• To establish a dichotomy in local growth rates independent of the Poisson intensity when the underlying motion is a general diffusion.
• To derive sharp asymptotics for the quenched global growth rate when dimension d ≤ 2 and for the annealed global growth rate in any dimension d.
• To identify subexponential correction terms in the growth rate asymptotics, providing quantitative precision beyond qualitative behavior.

Proposed method

• Models the system as a strictly dyadic branching Brownian motion with spatially dependent branching rate β on the complement of K, where K is the a-neighborhood of a Poisson point process on ℝᵈ.
• Uses a coupling argument: the BBM with mild obstacles is equivalent in law to a 'randomly trimmed' version of the free BBM, where branches are deleted at random when they hit K.
• Applies results from Yule processes and coalescence times to compute the distribution of the most recent common ancestor's death time, crucial for deriving growth asymptotics.
• Employs large deviation techniques and moment estimates to control the tail behavior of the total mass process |Z|ₜ under quenched and annealed measures.
• Derives exact expressions for the quenched and annealed growth rates using pathwise analysis and moment generating function bounds.
• Uses an auxiliary result (Appendix) on the distribution of the coalescence time of two lineages in a Yule process, derived via backward time genealogy and negative binomial embedding.

Experimental results

Research questions

• RQ1 What is the quenched local growth rate of branching Brownian motion when branching is suppressed in random Poissonian obstacle regions?
• RQ2 How does the global growth rate behave asymptotically under quenched and annealed measures, and what are the subexponential correction terms?
• RQ3 Does the dichotomy in local growth rates persist when the underlying motion is an arbitrary diffusion process, independent of the Poisson intensity?
• RQ4 Can precise asymptotics for the total mass process be derived in the quenched regime for d ≤ 2 and in the annealed regime for any d?
• RQ5 What is the distribution of the coalescence time of two ancestral lineages in a Yule process conditioned on survival to time t, and how does it relate to the growth rate?

Key findings

• The quenched local growth rate of the BBM with mild obstacles is equal to the branching rate β in the free region Kᶜ, independent of the obstacle intensity ν.
• For d ≤ 2, the quenched global growth rate satisfies |Z|ₜ ≍ exp(βt - c₁t^{1/2} + o(t^{1/2})) in probability, with an explicit subexponential correction term.
• For any d ≥ 1, the annealed global growth rate satisfies 𝔼[|Z|ₜ] ≍ exp(βt - c₂t^{1/2} + o(t^{1/2})) in probability, with a similar subexponential correction.
• The asymptotics for both quenched and annealed global growth are sharp and exhibit the same subexponential correction term, indicating a universal behavior in the growth rate.
• The distribution of the coalescence time of two ancestral lineages in a Yule process is derived explicitly, with a density function involving exponential and rational terms.
• The derived bound (45) in the appendix provides a precise formula for the coalescence time density, which is instrumental in proving the main asymptotic results.