CONVERGENCE OF MARKOV CHAIN APPROXIMATIONS TO STOCHASTIC REACTION DIFFUSION EQUATIONS

Michael A. Kouritzin () and Hongwei Long ()
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Michael A. Kouritzin: Department of Mathematical Sciences, University of Alberta
Hongwei Long: Department of Mathematical Sciences, University of Alberta

No lrsp-TRS361, RePAd Working Paper Series from Département des sciences administratives, UQO

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 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.

JEL-codes: C10 C40 (search for similar items in EconPapers)
Date: Written 2001-01-01
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