[Paper Review] A Branching-selection process related to censored Galton-Walton processes

Olivier Couronné, Lucas Gerin|arXiv (Cornell University)|Nov 4, 2011

Stochastic processes and statistical mechanics3 references4 citations

TL;DR

This paper establishes the asymptotic speed of a branching-selection particle system by linking it to a censored supercritical Galton-Watson process. It proves that the expected survival time of the censored process grows as $ (1/q)^N $, leading to tight bounds on the asymptotic speed $ v_N $, which converges to 1 exponentially fast in $ N $, with $ 1 - v_N \sim q^N $. This connection enables precise characterization of the system's propagation speed.

ABSTRACT

We obtain the asymptotics for the speed of a particular case of a particle system with branching and selection introduced by B\\'erard and Gou\\'er\\'e (2010). The proof is based on a connection with a supercritical Galton-Watson process censored at a certain level.

Motivation & Objective

• To analyze the asymptotic speed of a discrete-time branching-selection particle system with $ N $ particles on $ \mathbb{Z} $, where particles branch and selection retains only the $ N $ rightmost particles.
• To establish rigorous bounds on the asymptotic speed $ v_N $, particularly in the limit as $ N \to \infty $, for a class of branching mechanisms with $ \mathbb{E}[\mathcal{X}] > 1 $.
• To connect the particle system's behavior to a censored Galton-Watson process, enabling the use of known extinction time asymptotics for speed estimation.
• To provide a refined analysis of the survival time $ U_N $ of the censored Galton-Watson process, showing $ \mathbb{E}[U_N] \sim (1/q)^N $, which directly informs the speed bounds.

Proposed method

• Define a particle system where each of $ N $ particles at position $ \ell $ produces $ \mathcal{X} $ particles at $ \ell+1 $ and $ \mathcal{X}' $ at $ \ell $, then selects the $ N $ rightmost particles.
• Introduce a dominated coupling process $ \tilde{Y}_k $ that restarts with all particles at the rightmost position after each 'reset' time, ensuring stochastic domination over the original process.
• Define renewal times $ \Gamma^i $ based on the times $ V^i $ when all $ N $ particles are at the rightmost possible position, forming a regenerative structure.
• Use the renewal reward theorem to relate the expected number of renewals $ I_k $ by time $ k $ to the expected time between renewals, yielding $ \lim_{k\to\infty} \frac{1}{k}\mathbb{E}[I_k] = \frac{1}{\mathbb{E}[V^1]+1} $.
• Establish that $ \max Y_k \geq \max \tilde{Y}_k = k - I_k $, so $ \frac{1}{k}\mathbb{E}[\max Y_k] \geq 1 - \frac{1}{\mathbb{E}[V^1]+1} $, leading to a lower bound on $ v_N $.
• Leverage the connection between $ V^1 $ and the survival time $ U_N $ of a censored Galton-Watson process to show $ \mathbb{E}[V_N] = \mathbb{E}[U_N] $, enabling use of known asymptotics $ \mathbb{E}[U_N] \sim (1/q)^N $.

Experimental results

Research questions

• RQ1 What is the asymptotic speed $ v_N $ of the branching-selection particle system as $ N \to \infty $?
• RQ2 How does the survival time of a censored Galton-Watson process relate to the propagation speed in the particle system?
• RQ3 Can the speed $ v_N $ be bounded using a coupling with a regenerative process derived from the censored Galton-Watson process?
• RQ4 What is the precise asymptotic behavior of $ \mathbb{E}[U_N] $, the expected survival time of the censored Galton-Watson process with absorption at level $ N $?
• RQ5 How does the exponential decay of $ 1 - v_N $ in $ N $ emerge from the underlying branching and selection dynamics?

Key findings

• The asymptotic speed $ v_N $ of the particle system satisfies $ 1 - v_N \sim q^N $ as $ N \to \infty $, where $ q $ is the extinction probability of the underlying Galton-Watson process.
• The expected survival time $ \mathbb{E}[U_N] $ of the censored Galton-Watson process grows as $ (1/q)^N $, which is the key input for bounding the speed $ v_N $.
• The lower bound on $ v_N $ is given by $ v_N \geq 1 - \frac{1}{\mathbb{E}[V_N] + 1} $, and the upper bound by $ v_N \leq 1 - \frac{1}{\mathbb{E}[U_N]} $, both of which are asymptotically equivalent to $ 1 - q^N $.
• The random variable $ V^1 $, representing the time until all particles are at the rightmost position, has the same distribution as the survival time $ U_N $ of the censored Galton-Watson process.
• The coupling with the dominated process $ \tilde{Y}_k $ ensures $ \max Y_k \geq \max \tilde{Y}_k = k - I_k $, and the renewal structure yields the asymptotic speed estimate via the renewal theorem.
• The result confirms that the speed $ v_N \to 1 $ exponentially fast in $ N $, with the rate governed by the extinction probability $ q $ of the offspring distribution.