Convergence of Markov chain approximations to stochastic reaction diffusion equations.

• Author(s) / Creator(s)

  • Kouritzin, Michael
  • Long, H.

• 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

• Subjects / Keywords

  • Markov chains
  • Annealed law of large numbers
  • Quenched law of large numbers
  • Poisson processes
  • Stochastic reaction-diffusion equations

• Type of Item

  Article (Published)

• DOI

  https://doi.org/10.7939/R3MK65C50

• License

  Attribution-NonCommercial-NoDerivatives 3.0 International