Path integral solution of the Li-Chen equation and its application to global dynamic reliability via time-variant extreme process

Accurate quantification of probabilistic responses and global dynamic reliability of complex nonlinear stochastic systems remains a central challenge in engineering mechanics. Among existing methods, the probability density evolution method (PDEM) has shown significant promise due to its physical interpretability, computational efficiency, and nonintrusive implementation. This study advances PDEM by establishing a novel integral form of the Li-Chen equation - the fundamental governing equation derived from the principle of probability preservation. Specifically, the Li-Chen equation is reformulated within the path integral solution (PIS) framework, offering enhanced numerical stability while maintaining mathematical equivalence to its partial differential form. Based on this framework, an accurate and efficient numerical solving procedure is developed and implemented for stochastic systems. Furthermore, the proposed PIS framework addresses a longstanding challenge in system-level dynamic reliability analysis: capturing time-variant, multi-mode failure dependencies without incurring excessive computational costs. By coupling the integral-form Li-Chen equation with a time-variant extreme process model, a new methodology is developed for evaluating global dynamic reliability in systems subject to multiple correlated failure modes. This method overcomes the limitations in existing approaches that often require repeated simulations at different thresholds and time instants. To demonstrate the accuracy, robustness, and computational efficiency of the proposed method, numerical examples involving complex nonlinear systems are presented. This work provides both a theoretical advancement in PDEM formulation and a practical tool for system-level reliability assessment under stochastic dynamic excitations.