A Functional Central Limit Theorem for the Becker–Döring Model

Abstract

We investigate the fluctuations of the stochastic Becker-Döring 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.

On the Asymptotic Distribution of Nucleation Times of Polymerization Processes

“…and, since the coordinates of the last process are upper bounded by ρt/N , we deduce the convergence to (0) of the sequence of processes (31). The previsible increasing previsible process of the corresponding martingale is given by…”

“…In a stochastic context, Jeon [16] shows for the Smoluchowski model, that with Poisson processes governing the dynamics of the transitions of binary coagulation and fragmentation, then, under appropriate conditions, a convergence result holds for the coordinates properly scaled and the limit is the solution of a set of deterministic ODE's. For this model the corresponding functional central limit theorem has been proved in Sun [31]. See Szavits et al [33], Eugène et al [11] and Doumic et al [9] for related stochastic models.…”

The Becker–Döring Process: Pathwise Convergence and Phase Transition Phenomena

“…The focus of the work by Jeon was on more general coagulationfragmentation models though, and on gelling solutions (that may arise in finite time for some coagulation-fragmentation models). The second work is by Sun in [18], who proves a strong law of large numbers (in the spirit of Kurtz theorem) using bounded kinetics rates. In such case, the right-hand side of the DBD system (1) is clearly Lipschitz on X.…”

Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

“…We refer to our review [4] for historical comments and detailed literature review on theoretical results from the deterministic side. See also [5,10] for recent results on functional law of large number and central limit theorem.…”