动力学福克-普朗克方程的核公式
A Kernel Formula for Kinetic Fokker-Planck Equations
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中文总结 AI 辅助
该研究推导了动力学福克-普朗克方程的显式单步核公式，建立了其局部弱一致性等性质，给出了变分解释并通过数值实验验证了其精度与应用价值。
中文摘要 AI 辅助
我们推导了动力学福克-普朗克方程的显式单步核公式。该构造从一个具有显式转移核的参考动力学开始，使用精心选择的辅助权重来编码漂移扰动。所得核公式提供了密度演化的显式近似。在加权相空间设定中，我们在适当的正则性和矩假设下建立了局部弱一致性和一阶有限时间弱收敛。我们还根据正则化瓦瑟斯坦型邻近算子对该公式给出了形式变分解释，并为更广泛的抛物型方程给出了类似构造。数值实验说明了核近似的精度及其在动力学采样的确定性粒子格式中的应用。
英文摘要
We derive an explicit one-step kernel formula for the kinetic Fokker-Planck equation. The construction begins with a reference dynamics admitting an explicit transition kernel and uses a carefully chosen auxiliary weight to encode the drift perturbation. The resulting kernel formula provides an explicit approximation of the density evolution. In a weighted phase-space setting, we establish local weak consistency and first-order finite-time weak convergence under suitable regularity and moment assumptions. We also give a formal variational interpretation of the formula in terms of a regularized Wasserstein-type proximal operator and present analogous constructions for a broader class of parabolic equations. Numerical experiments illustrate the accuracy of the kernel approximation and its application to deterministic particle schemes for kinetic sampling.
