A Functional Central Limit Theorem for the Becker-D\"oring model

TL;DR

This paper establishes a functional central limit theorem for the stochastic Becker-D"oring model of polymerization, showing convergence to an infinite-dimensional SDE and characterizing equilibrium fluctuations as Gaussian processes.

Contribution

It introduces a novel functional CLT for the model, linking stochastic fluctuations to an infinite-dimensional SDE framework and analyzing equilibrium Gaussian behavior.

Findings

Fluctuations converge to an infinite-dimensional SDE

At equilibrium, the process is Gaussian

Provides technical estimates for fluctuation control

Abstract

We investigate the fluctuations of the stochastic Becker-D\"oring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the vector of the number of polymers of a given size. It is shown that the stochastic process associated to fluctuations is converging to the strong solution of an infinite dimensional stochastic differential equation (SDE) in a Hilbert space. We also prove that, at equilibrium, the solution of this SDE is a Gaussian process. The proofs are based on a specific representation of the evolution equations, the introduction of a convenient Hilbert space and several technical estimates to control the fluctuations, especially of the first coordinate which interacts with all components of the infinite dimensional vector representing the state of the process.