Given two distinct subsets A, B in the state space of some dynamical system, transition path theory (TPT) was successfully used to describe the statistical behavior of transitions from A to B in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit and provide this powerful tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. This is partially motivated by studying applications such as climate, ocean, and social dynamics. On simple model examples, we show how the new tools are able to deliver quantitative understanding about the statistical behavior of such systems. We also point out explicit cases where the more general dynamical regimes show different behaviors to their stationary counterparts, linking these tools directly to bifurcations in non-deterministic systems.

给定某些动力学系统的状态空间中的两个不同子集A，  B，转换路径理论（TPT）成功地用于描述从A到B的转换的统计行为在固定系统的遍历极限内。我们推导出了TPT的概括，它消除了平稳性和遍历极限的要求，并为分析其他动力学场景提供了强大的工具：周期性强制动力学和与时间相关的有限时间系统。这是通过研究诸如气候，海洋和社会动态等应用程序而部分激发的。在简单的模型示例中，我们展示了新工具如何能够对此类系统的统计行为提供定量的理解。我们还指出了明显的情况，在这些情况下，更一般的动力体制与固定动力体制表现出不同的行为，将这些工具直接链接到非确定性系统中的分支。