Quantitative Propagation of Chaos in a Bimolecular Chemical Reaction-Diffusion Model
Quantitative Propagation of Chaos in a Bimolecular Chemical Reaction-Diffusion Model
复制标题
双分子化学反应扩散模型中混沌的定量传播
DOI：
10.1137/19m1287687
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
J. Nolen
中科院分区：
文献类型：
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作者：
Tau Shean Lim;Yulong Lu;J. Nolen
我们研究了一个随机系统的$N$相互作用的粒子模型双分子化学反应扩散。在这个模型中，每个粒子$i$带有两个属性：空间位置$X_t^i\in \mathbb{T}^d$和类型$\Xi_t^i\in \{1，\cdots，n\}$。虽然$X_t^i$是一个标准的（独立的）扩散过程，但类型$\Xi_t^i$的演化是通过化学反应网络描述的一系列化学反应下不同粒子之间的成对相互作用来描述的。我们证明了在大粒子极限下，随机动力学收敛到一个由非局部反应扩散偏微分方程描述的平均场极限。特别地，我们得到了相互作用粒子系统混沌传播的定量结果。我们的证明是基于Jabin和Wang \cite{JW 18}最近使用的相对熵方法。相对熵方法的关键成分是一个特殊的配分函数的大偏差估计，这是以前证明的技术组合估计。我们给出了一个简单的概率证明的基础上一个新的鞅的论点。
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 $X_t^i\in \mathbb{T}^d$, and the type $\Xi_t^i\in \{1,\cdots,n\}$. While $X_t^i$ is a standard (independent) diffusion process, the evolution of the type $\Xi_t^i$ 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 \cite{JW18}. 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.