论文标题
随机矢量束的动力学方程
Kinetic Equation for Stochastic Vector Bundles
论文作者
论文摘要
动力学方程对于理解随机过程的统计特性至关重要，但是电流方程（例如经典的Fokker-Planck）仅限于局部分析。本文为矢量束上的随机系统提供了一个新的动力学方程，以解决全球尺度随机性。动力学方程是通过集合平均局部概率密度函数的累积扩展得出的，该局部概率密度函数是状态过渡轨迹的功能。动力学方程是概率空间的测量方程。它捕获了全球和历史影响，说明了非马克维亚性，并可以简化为马尔可夫流程的古典福克 - 普兰克方程。本文还讨论了有关动力学方程式的相对问题，包括非马尔可夫，马尔可夫近似，宏观保护方程，量规转换以及无限级动力学方程的截断以及需要进一步关注的限制。
The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker-Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker-Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.
