论文标题
McKean-Vlasov PDE具有不规则漂移，并应用于保守的SPDE的大偏差
McKean-Vlasov PDE with Irregular Drift and Applications to Large Deviations for Conservative SPDEs
论文作者
论文摘要
受[Fehrman，Gess的启发；发明。 Math。，2023]，我们对McKean-Vlasov PDE进行了精细的分析，具有单数相互作用和平方根形式的漂移项。作为与单数相互作用（随机，保守的PDE）的院长卡瓦萨基方程的相应骨架方程，它决定了小噪声大偏差的速率函数。通过在相互作用内核上施加Ladyzhenskaya-serrodi-serrin型条件，当噪声的强度和相关性同时发送到合适的缩放量表下的强度和相关性时，我们就会在随机重新归一化的动力学解决方案的框架中建立较大的偏差。该结果有助于证明与奇异相互作用的平均场系统相关的宏观波动理论与与Dean-Kawasaki方程相关的流体动力学相关的一致性。作为一种应用，我们还获得了其他随机保守的PDE的巨大偏差，称为波动ising-kac-kawasaki动力学。这对于探索川崎动态伊辛 - 卡克模型的波动非常重要，因为它们正式在高斯波动，较大的偏差和接近临界的限制方面表现出相同的关键特征。
Inspired by [Fehrman, Gess; Invent. Math., 2023], we provide a fine analysis of the McKean-Vlasov PDE with singular interactions and drift terms of square root form. As the corresponding skeleton equation of Dean-Kawasaki equation with singular interactions (a stochastic, conservative PDE), it determines the rate function of small noise large deviations. By imposing Ladyzhenskaya-Prodi-Serrin type conditions on the interaction kernel, we establish the large deviations in the framework of stochastic renormalized kinetic solution, when the intensity and the correlation of the noise are simultaneously sent to $0$ under a suitable scaling. This result contributes to demonstrating the consistency between the macroscopic fluctuation theory associated with singular interacting mean-field systems and fluctuating hydrodynamics related to the Dean-Kawasaki equation. As an application, we also obtain large deviations for the other stochastic conservative PDE called fluctuating Ising-Kac-Kawasaki dynamics. It is of great importance in exploring fluctuations of Kawasaki dynamical Ising-Kac model, since they formally exhibits the same key features in terms of Gaussian fluctuations, large deviations, and scaling limits near criticality.
