Transition path theory (TPT) for diffusion processes is a framework for analysing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretisation of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretisation. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.

中文翻译：

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扩散过程过渡路径理论的收敛离散化方法

扩散过程的转移路径理论 (TPT) 是一个框架，用于分析不相交的状态空间亚稳态子集之间的多尺度遍历扩散过程的转移。大多数应用 TPT 的方法都涉及在近似潜在扩散过程的状态空间离散化上构建马尔可夫状态模型。然而，马尔可夫假设在实践中难以验证，迄今为止，这些方法还没有已知的误差界限或收敛结果。我们提出了一种蒙特卡罗方法，用于逼近 TPT 的前向提交者、概率电流和流线，用于扩散过程。我们的方法仅使用样本轨迹数据和基于 Voronoi 镶嵌的状态空间分区。它不需要构建马尔可夫逼近过程。我们严格证明了近似 TPT 对象的误差界限，并使用这些界限来显示在任意精细离散化的限制下与其精确对应物的收敛。我们通过应用于在三阱电位上求解 Smoluchowski 方程的过程来说明我们方法的一些特征。