  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.
