Dimension-reduced Chapman-Kolmogorov equation for high-dimensional stochastic dynamical systems

Random vibration analysis of high-dimensional dynamical systems is a fundamental problem in science and engineering, yet it remains challenging due to the curse of dimensionality. While dimension-reduced formulations have been developed for differential-type equations governing time-variant probability density, such as the Fokker-Planck equation, no equivalent formulation has been established for the integral-type Chapman-Kolmogorov (CK) equation, despite its theoretical importance and computational advantages. In this paper, a novel dimension-reduced Chapman-Kolmogorov (DRCK) equation is established governing the transient probability density function (PDF) of any quantity of interest in high-dimensional Markov systems. The derivation is conducted based on the projection of the full Chapman-Kolmogorov equation onto the dimension-reduced space. It is established that the intrinsic transition probability density (TPD) of the DRCK equation is the conditional expectation of the original TPD. Further, the short-time approximate intrinsic TPDs under both Gaussian and Poisson white noise excitations are derived analytically, enabling practical numerical implementation. The proposed DRCK equation provides a mathematically rigorous and computationally efficient framework for high-dimensional stochastic systems. Numerical examples are developed to demonstrate its accuracy and effectiveness. The DRCK equation thus provides a new tool for reliability assessment and uncertainty quantification in complex engineering applications.