Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the $\textit{Ehrenfest process}$, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.

翻译：基于随机过程的生成建模不仅在实证上取得了显著成果，也推动了其理论理解的最新进展。原则上，过程的空间和时间维度均可为离散或连续形式。本研究聚焦于离散状态空间上的时间连续马尔可夫跳跃过程，并探讨其与由随机微分方程给出的状态连续扩散过程之间的对应关系。特别地，我们重新审视了**埃伦费斯特过程**，该过程在无限状态空间极限下收敛至奥恩斯坦-乌伦贝克过程。同样地，我们证明了埃伦费斯特过程的时间反转过程收敛至时间反转的奥恩斯坦-乌伦贝克过程。这一发现弥合了离散与连续状态空间之间的鸿沟，使得方法可在两种设定间相互迁移。此外，我们提出了一种基于条件期望的马尔可夫跳跃过程时间反转训练算法，该算法可直接与去噪分数匹配相关联。我们通过多个具有说服力的数值实验验证了所提方法的有效性。