Convergence of Markov chain approximations to stochastic reaction-diffusion equations

• 成果类型: Article
• 署名作者: Kouritzin, MA; Long, HW
• 署名单位: University of Alberta
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 1039-1070
• 关键词: finite-element method; contaminant transport; porous-media; nonequilibrium adsorption; chemical-reactions; MODEL; groundwater; limit; biodegradation; solutes

摘要

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-hundredfold 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.