A law of large numbers and a central limit theorem are proved for a locally interacting particle system. This system describes a chemical reaction with diffusion with linear creation and quadratic annihilation of particles. The deterministic limit is the solution of a nonlinear reaction-diffusion equation defined on an $n$-dimensional unit cube. The law of large numbers holds for any dimension $n$ and arbitrary times, whereas the central limit theorem holds only for dimension $n \leq 3$ and on a certain bounded time interval (depending on the initial distribution and on the creation rate). A propagation of chaos expansion of the correlation functions is used.
Publié le : 1992-08-14
Classification:
Nonlinear reaction-diffusion equation,
locally interacting particle system,
thermodynamic limit,
Gaussian limit,
van Kampen's approximation,
propagation of chaos,
BBGKY hierearchy,
stochastic evolution equations,
spatially inhomogeneous population growth,
60K35,
60G57,
60H15,
35K55,
60F17,
60J70
@article{1177005654,
author = {Kotelenez, Peter},
title = {Fluctuations in a Nonlinear Reaction-Diffusion Model},
journal = {Ann. Appl. Probab.},
volume = {2},
number = {4},
year = {1992},
pages = { 669-694},
language = {en},
url = {http://dml.mathdoc.fr/item/1177005654}
}
Kotelenez, Peter. Fluctuations in a Nonlinear Reaction-Diffusion Model. Ann. Appl. Probab., Tome 2 (1992) no. 4, pp. 669-694. http://gdmltest.u-ga.fr/item/1177005654/