## 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^{i} _{t} ∈ T^{d}, and the type [I]^{i} _{t} ∈ { 1, . . ., n} . While X^{i} _{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.

Original language | English (US) |
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Pages (from-to) | 2098-2133 |

Number of pages | 36 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

### Bibliographical note

Funding Information:\ast Received by the editors September 16, 2019; accepted for publication (in revised form) January 21, 2020; published electronically April 28, 2020. https://doi.org/10.1137/19M1287687 Funding: The work of James Nolen was partially funded through grant DMS-1351653 from the National Science Foundation. \dagger Department of Mathematics, Duke University, Durham, NC 27708 (taushean@math.duke.edu, yulonglu@math.duke.edu, nolen@math.duke.edu).

Publisher Copyright:

© 2020 Society for Industrial and Applied Mathematics.

## Keywords

- Hydrodynamics limit
- Interacting particle systems
- Large deviations
- Propagation of chaos
- Reaction-diffusion system