Comparison of Stochastic and Deterministic Models of a Linear Chemical Reaction with Diffusion

Abstract

Particles placed in $N$ cells on the unit interval give birth or die according to linear rates. Adjacent cells are coupled by diffusion with a rate proportional to $N^2$. Cell numbers are divided by a density parameter to represent concentrations, and the resulting space-time Markov process is compared to a corresponding deterministic model, the solution to a partial differential equation. The models are viewed as Hilbert space valued processes and compared by means of a law of large numbers and central limit theorem. New and nearly optimal results are obtained by exploiting the Ornstein-Uhlenbeck type structure of the stochastic model.