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August 2002 Convergence of Markov chain approximations to stochastic reaction-diffusion equations
Michael A. Kouritzin, Hongwei Long
Ann. Appl. Probab. 12(3): 1039-1070 (August 2002). DOI: 10.1214/aoap/1031863180

Abstract

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.

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Michael A. Kouritzin. Hongwei Long. "Convergence of Markov chain approximations to stochastic reaction-diffusion equations." Ann. Appl. Probab. 12 (3) 1039 - 1070, August 2002. https://doi.org/10.1214/aoap/1031863180

Information

Published: August 2002
First available in Project Euclid: 12 September 2002

zbMATH: 1018.60063
MathSciNet: MR1925451
Digital Object Identifier: 10.1214/aoap/1031863180

Subjects:
Primary: 60H15 , 60K35
Secondary: 60F15 , 60G55 , 60J27

Keywords: annealed law of large numbers , Markov chains , Poisson processes , quenched law of large numbers , Stochastic reaction-diffusion equations

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.12 • No. 3 • August 2002
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