路径空间上的熵及其在奇异扩散和平均场模型中的应用
Entropy on the path space and application to singular diffusions and mean-field models
AI总结：
本文基于路径空间相对熵提出奇异相互作用粒子系统及其McKean-Vlasov极限的统一处理方法，推导了多类奇异扩散的存在唯一性，证明了粒子数趋于无穷时的收敛性与混沌传播，给出已知结果的新证明、拓展及全新结论。
摘要：
本文旨在对各类奇异相互作用粒子系统及其McKean-Vlasov（麦克恩-弗拉索夫）极限提出统一的处理方法。该统一方法基于路径空间上相对熵的运用，遵循我们此前与C. Léonard合作研究的思路。我们展示了该方法可用于推导若干奇异扩散的存在性与唯一性，尤其是线性平均场随机粒子系统和McKean-Vlasov型非线性随机微分方程（SDE），涵盖$\boldsymbol{L}^p$-$\boldsymbol{L}^q$模型、与二维Navier-Stokes（纳维-斯托克斯）方程相关的二维涡旋模型、次库仑相互作用模型以及Patlak-Keller-Segel（帕特拉克-凯勒-西格尔）模型。我们还证明了当粒子数趋于无穷时的收敛性与混沌传播特性。该结论主要在过程层面得到，而非仅在刘维尔方程（边缘分布流）层面。因此，本文包含已知结果的新证明与拓展，也包含全新结果，主要结论置于引言末尾。
英文摘要：
In this paper we intend to present a unified treatment of a variety of singular interacting particle systems and their McKean-Vlasov limits. This unified approach is based on the use of the relative entropy on the path space in the spirit of our previous works together with C. L{é}onard. We show how it can be used to derive existence and uniqueness for some singular diffusions, in particular linear mean field stochastic particle systems and non linear SDE of McKean-Vlasov type, including $\mathbf L^p-\mathbf L^q$ models, the 2D vortex model associated to the 2D Navier-Stokes equation, sub-Coulombic interactions models or the Patlak-Keller-Segel model. We also show the convergence and propagation of chaos as the number of particles grows to infinity. This is (mainly) obtained at the process level, not only at the Liouville equation (marginals flow) level. The paper thus contains new proofs and extensions of known results, as well as new results.The main results are given at the end of the Introduction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
