Quantitative Propagation of Chaos in the bimolecular chemical reaction-diffusion model

2019/06/03 by Lim, Tau Shean, Lu, Yulong, Nolen, James · 1 citation
#Analysis of PDEs #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1906.01051

Abstract

We study a stochastic system of N interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle i carries two attributes: the spatial location X_tⁱ∈ \mathbbT^d, and the type Ξ_tⁱ∈ \1,⋯,n\. While X_tⁱ is a standard (independent) diffusion process, the evolution of the type Ξ_tⁱ is described by pairwise interactions between different particles under a series of chemical reactions described by a chemical reaction network. We prove that in the large particle limit the stochastic dynamics converges to a mean field limit which is described by a nonlocal reaction-diffusion partial differential equation. In particular, we obtain a quantitative propagation of chaos result for the interacting particle system. Our proof is based on the relative entropy method used recently by Jabin and Wang \citeJW18. The key ingredient of the relative entropy method is a large deviation estimate for a special partition function, which was proved previously by technical combinatorial estimates. We give a simple probabilistic proof based on a novel martingale argument.

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