Convergence of Markov chain approximations to stochastic reaction diffusion equations.

  • Author(s) / Creator(s)
  • 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.

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  • Type of Item
    Article (Published)
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    Attribution-NonCommercial-NoDerivatives 3.0 International
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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.