Quantitative propagation of chaos in a bimolecular chemical reaction-diffusion model

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 Xⁱ _t ∈ T^d, and the type [I]ⁱ _t ∈ { 1, . . ., n} . While Xⁱ _t is a standard (independent) diffusion process, the evolution of the type [I]ⁱ _t is described by pairwise interactions between different particles under a series of chemical reactions described by a chemical reaction network. We prove that, as N → ∞, the stochastic system has 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 [Invent. Math., 214 (2018), pp. 523-591]. The key ingredient of the relative entropy method is a large deviation estimate for a special partition function, which was proved previously by combinatorial estimates. We give a simple probabilistic proof based on a novel martingale argument.