[Paper Review] F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system

Pablo Groisman, Matthieu Jonckheere+1 more|arXiv (Cornell University)|Mar 29, 2019

Stochastic processes and statistical mechanics25 references4 citations

TL;DR

This paper studies a continuous-time particle system where N particles perform independent Brownian motion and undergo pairwise branching-selection events, with the rightmost particle in each pair replacing both. It proves that the empirical measure's cumulative distribution function converges to a solution of the F-KPP equation, and establishes that the system's macroscopic speed converges to the minimal traveling wave speed √2 as N → ∞, confirming a selection principle for Brunet-Derrida-type models via hydrodynamic limits and coupling arguments.

ABSTRACT

We study a particle system with the following diffusion-branching-selection mechanism. Particles perform independent one dimensional Brownian motions and on top of that, at a constant rate, a pair of particles is chosen uniformly at random and both particles adopt the position of the rightmost one among them. We show that the cumulative distribution function of the empirical measure converges to a solution of the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation and use this fact to prove that the system selects the minimal macroscopic speed as the number of particles goes to infinity.

Motivation & Objective

• To establish a rigorous hydrodynamic limit for a finite-population particle system with diffusion, branching, and selection dynamics.
• To prove that the asymptotic speed of the system converges to the minimal speed of the F-KPP equation as the number of particles N → ∞.
• To provide a non-perturbative, comparison-based proof of the speed selection principle without relying on Legendre transforms or explicit computations.
• To quantify propagation of chaos via explicit decorrelation bounds in the system.
• To demonstrate that the limiting behavior is exactly described by the F-KPP equation, not an approximation.

Proposed method

• The system is defined with N particles performing independent one-dimensional Brownian motions, and at Poisson times, a random pair is selected and both particles adopt the position of the rightmost one.
• The cumulative distribution function of the empirical measure is shown to converge to a solution of the F-KPP equation ∂tu = ½∂²xu + u² − u via propagation of chaos arguments.
• A stationary regime is established for the system seen from the leftmost particle, enabling the definition of a finite-N velocity v_N.
• The lower bound on v_N is derived using convergence to the F-KPP solution and a comparison argument involving the integral of F_N(1−F_N).
• The upper bound is obtained by coupling the system with N independent Branching Brownian Motions (BBMs), showing that the system's maximum is stochastically dominated by the maximum of N i.i.d. BBMs.
• The asymptotic speed is bounded above by √2 using the known almost sure limit lim_{t→∞} M_t / t = √2 for a BBM.

Experimental results

Research questions

• RQ1 Does the empirical measure of a continuous-time particle system with branching and selection converge to a solution of the F-KPP equation in the hydrodynamic limit?
• RQ2 Does the system select the minimal traveling wave speed of the F-KPP equation as the number of particles N → ∞?
• RQ3 Can the speed selection principle be rigorously proven without relying on Legendre transforms or explicit moment calculations?
• RQ4 How can propagation of chaos be quantified in such a system with pairwise selection events?
• RQ5 Can the limiting behavior be exactly described by the F-KPP equation, rather than an approximation?

Key findings

• The cumulative distribution function of the empirical measure converges in law to a solution of the F-KPP equation as N → ∞.
• The system's macroscopic speed converges to √2, the minimal speed of the F-KPP equation, as N → ∞.
• The lower bound for the finite-N velocity v_N is established via convergence of F_N(1−F_N) to the F-KPP solution and a probabilistic comparison argument.
• The upper bound is derived by coupling the system with N independent BBMs, showing that the maximum of the system is stochastically dominated by the maximum of the BBMs.
• The asymptotic speed of the system satisfies liminf_{N→∞} v_N ≥ √2 and limsup_{N→∞} v_N ≤ √2, hence lim_{N→∞} v_N = √2.
• The analysis provides explicit decorrelation bounds, enabling a quantitative control of propagation of chaos in the system.