Bridging discrete and continuous state spaces: Exploring the Ehrenfest process in time-continuous diffusion models

While the original attempt considers processes can be discrete or continuous. In this discrete-time, continuous-space processes (Ho et al., work, we study time-continuous Markov jump 2020), one can show that in the small step-size limit the processes on discrete state spaces and investigate models converge to continuous-time, continuous-space processes their correspondence to state-continuous diffusion given by stochastic differential equations (SDEs) processes given by SDEs. In particular, we revisit (Song et al., 2021). This continuous time framework then the Ehrenfest process, which converges to an allows fruitful connections to mathematical tools such as Ornstein-Uhlenbeck process in the infinite state partial differential equations, path space measures and optimal space limit. Likewise, we can show that the timereversal control (Berner et al., 2024). As an alternative, one of the Ehrenfest process converges to the can consider discrete state spaces in continuous time via time-reversed Ornstein-Uhlenbeck process. This Markov jump processes, which have been suggested for observation bridges discrete and continuous state generative modeling in Campbell et al. (2022). Those are spaces and allows to carry over methods from one particularly promising for problems that naturally operate to the respective other setting. Additionally, we on discrete data, such as, e.g., text, images, graph structures suggest an algorithm for training the time-reversal or certain biological data, to name just a few.