# A Complete Decomposition of Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2601.07834 · HTML: https://mlsanity.com/paper/2601.07834/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 10 of 10)

- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Ensemble preconditioning for Markov chain Monte Carlo simulation](https://mlsanity.com/paper/1607.03954/) (2016)
- [Scalable Bayesian Learning with posteriors](https://mlsanity.com/paper/2406.00104/) (2024)

## Built on by (most cited first, 1 of 1)

- [CN101 - A Digital Thermodynamic Computer for Generative AI](https://mlsanity.com/paper/2608.00754/) (2026)

## Abstract (arXiv)

We show that any stochastic differential equation with prescribed time-dependent marginal distributions admits a decomposition into three components: a unique scalar field governing marginal evolution, a symmetric positive-semidefinite diffusion matrix field and a skew-symmetric matrix field.