In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well-established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions. Our alterations also feature dramatically slower Markov chain state change rates often yielding a ten to one-hundred-fold simulation speed increase over the previous version of the method as evidenced in our computer implementations. On a weighted $L^{2}$ Hilbert space chosen to symmetrize the elliptic operator, we consider existence of and convergence to pathwise unique mild solutions of our stochastic reaction-diffusion equation. Our main convergence result, a quenched law of large numbers, establishes convergence in probability of our Markov chain approximations for each fixed path of our driving Poisson measure source. As a consequence, we also obtain the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.
Publié le : 2002-08-14
Classification:
Stochastic reaction-diffusion equations,
Markov chains,
Poisson processes,
quenched law of large numbers,
annealed law of large numbers,
60H15,
60K35,
60F15,
60J27,
60G55
@article{1031863180,
author = {Kouritzin, Michael A. and Long, Hongwei},
title = {Convergence of Markov chain approximations to stochastic reaction-diffusion equations},
journal = {Ann. Appl. Probab.},
volume = {12},
number = {1},
year = {2002},
pages = { 1039-1070},
language = {en},
url = {http://dml.mathdoc.fr/item/1031863180}
}
Kouritzin, Michael A.; Long, Hongwei. Convergence of Markov chain approximations to stochastic reaction-diffusion equations. Ann. Appl. Probab., Tome 12 (2002) no. 1, pp. 1039-1070. http://gdmltest.u-ga.fr/item/1031863180/