Connections between the Föllmer process and the denoising diffusion probabilistic model
Abstract
The Föllmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an ``augmented'' time-compressed version of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, the connection between direct discretization of the Föllmer process and the DDPM sampler has not yet been fully explored. This paper clarifies this point while surveying relevant results from the literature. We show that discretized Föllmer processes give natural hyper-parameter settings of the DDPM sampler while accommodating a broader class of variance schedules than discretized reverse SDEs. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds, along with slight improvements.
Related
- Spatio-temporal Latent Denoising Diffusion Probabilistic Models for Reduced-order Modeling of Parametrized Dynamical Systems
- RDDMPI: Residual Denoising Diffusion Model for Probabilistic Multivariate Time Series Imputation
- Convergence of Denoising Diffusion Probabilistic Models: A Rényi Divergence Perspective
- Schrödinger--Föllmer Actor--Critic: Diffusion Policy Improvement with Finite-Sample Analysis
- Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index
BibTeX
@misc{koike2026connections,
title = {{Connections between the Föllmer process and the denoising diffusion probabilistic model}},
author = {Yuta Koike},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.18040},
eprint = {2605.18040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.18040},
}
RIS
TY - UNPB
TI - Connections between the Föllmer process and the denoising diffusion probabilistic model
AU - Yuta Koike
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2605.18040
UR - https://arxiv.org/abs/2605.18040
AB - The Föllmer process is a Brownian motion conditioned to have a pre-specified distribution at time 1. This process can be interpreted as an ``augmented'' time-compressed version of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, the connection between direct discretization of the Föllmer process and the DDPM sampler has not yet been fully explored. This paper clarifies this point while surveying relevant results from the literature. We show that discretized Föllmer processes give natural hyper-parameter settings of the DDPM sampler while accommodating a broader class of variance schedules than discretized reverse SDEs. Moreover, this allows us to systematically recover state-of-the-art results on DDPM sampling error bounds, along with slight improvements.
ER -