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.

通过研究时态连续的马尔可夫跳跃过程和离散状态空间下的状态连续扩散过程之间的对应关系，本研究桥接了离散和连续状态空间，并提出了一种直接与去噪评分匹配相关的马尔可夫跳跃过程的时态逆过程训练算法。

离散和连续状态空间的桥梁：在时间连续的扩散模型中探索埃伦费斯特过程