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Permanent link (DOI): https://doi.org/10.7939/R3MK65C50

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Convergence of Markov chain approximations to stochastic reaction diffusion equations. Open Access

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Author or creator
Kouritzin, Michael
Long, H.
Additional contributors
Subject/Keyword
Markov chains
annealed law of large numbers
Stochastic reaction-diffusion equations
quenched law of large numbers
Poisson processes
Type of item
Journal Article (Published)
Language
English
Place
Time
Description
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 L2 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.
Date created
2002
DOI
doi:10.7939/R3MK65C50
License information
Creative Commons Attribution-Non-Commercial-No Derivatives 3.0 Unported
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Citation for previous publication
M.A. Kouritzin and H. Long, "Convergence of Markov chain approximations to stochastic reaction diffusion equations'', Annals of Applied Probability, 12, (2002), 1039-1070.
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