# Using Ornstein-Uhlenbeck Process to understand Denoising Diffusion Probabilistic Model and its Noise Schedules

arXiv: https://arxiv.org/abs/2311.17673 · HTML: https://mlsanity.com/paper/2311.17673/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 8 of its 11 references that are in the map.
- First posted on arXiv 29 Nov 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 8 of 8)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)

## Built on by (most cited first, 7 of 7)

- [The Cosine Schedule is Fisher-Rao-Optimal for Masked Discrete Diffusion Models](https://mlsanity.com/paper/2508.04884/) (2025)
- [Random Walks with Tweedie: A Unified View of Score-Based Diffusion Models](https://mlsanity.com/paper/2411.18702/) (2024)
- [On the Design of One-step Diffusion via Shortcutting Flow Paths](https://mlsanity.com/paper/2512.11831/) (2025)
- [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026)
- [Quantum-Inspired DRL Approach with LSTM and OU Noise for Cut Order Planning Optimization](https://mlsanity.com/paper/2508.16611/) (2025)
- [Let Physics Guide Your Protein Flows: Topology-aware Unfolding and Generation](https://mlsanity.com/paper/2509.25379/) (2025)
- [A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Models](https://mlsanity.com/paper/2601.13602/) (2026)

## Abstract (arXiv)

The aim of this short note is to show that Denoising Diffusion Probabilistic Model DDPM, a non-homogeneous discrete-time Markov process, can be represented by a time-homogeneous continuous-time Markov process observed at non-uniformly sampled discrete times. Surprisingly, this continuous-time Markov process is the well-known and well-studied Ornstein-Ohlenbeck (OU) process, which was developed in 1930's for studying Brownian particles in Harmonic potentials. We establish the formal equivalence between DDPM and the OU process using its analytical solution. We further demonstrate that the design problem of the noise scheduler for non-homogeneous DDPM is equivalent to designing observation times for the OU process. We present several heuristic designs for observation times based on principled quantities such as auto-variance and Fisher Information and connect them to ad hoc noise schedules for DDPM. Interestingly, we show that the Fisher-Information-motivated schedule corresponds exactly the cosine schedule, which was developed without any theoretical foundation but is the current state-of-the-art noise schedule.