AI 中文总结
研究从适度相互作用随机粒子系统推导多物种交叉扩散方程的混沌传播定量估计，用相对熵方法分两步证明\(L^1\)范数下结果，结合相关收敛与估计经插值得\(L^q\)结果。
AI 中文摘要
在本文中，我们证明了混沌传播的定量结果，该结果使我们能够从中度相互作用的随机粒子系统推导出多物种交叉扩散方程。通过相对熵方法获得了\(L^1\)范数下的混沌传播定量结果，证明分两步进行。第一步，我们在中间层面量化粒子系统的联合分布与偏微分方程（PDE）的张量化解之间的相对熵。第二步，通过分析中间层面PDE的解与极限PDE的解之间的\(L^2\)距离，建立到多物种交叉扩散方程的严格收敛速率。此外，结合混沌传播的强\(L^1\)收敛与多物种粒子系统边际分布的\(L^p\)估计（\(2\leq p<\infty\)），我们通过插值推导出相应的\(L^q\)结果（\(1<q<\infty\)）。
英文摘要
In this paper, we prove the quantitative propagation of chaos results that allow us to derive multi-species cross-diffusion equations from moderately interacting stochastic particle system. The quantitative propagation of chaos result in $L^1$-norm is obtained by the relative entropy method, and the proof is carried out in two steps. In the first step, we quantify the relative entropy between the joint distribution of the particle system and the tensorised solution of the PDE at the intermediate level. In the second step, we establish a rigorous convergence rate to the multi-species cross-diffusion equations by analyzing the $L^2$-distance between the solution of the intermediate-level PDE and that of the limiting PDE. Furthermore, combining the strong $L^1$-convergence for the propagation of chaos with the $L^p$-estimates $(2\le p<\infty)$ for the marginal distribution of multi-species particle system, we derive the corresponding $L^q$-result $(1<q<\infty)$ via interpolation.
