# Malliavin Calculus for Score-based Diffusion Models

arXiv: https://arxiv.org/abs/2503.16917 · HTML: https://mlsanity.com/paper/2503.16917/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 21 Mar 2025.

## Placed in

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

## Builds on (most cited first, 6 of 6)

- [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)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation](https://mlsanity.com/paper/2210.04296/) (2022)

## Built on by (most cited first, 6 of 6)

- [Conditioning Diffusions Using Malliavin Calculus](https://mlsanity.com/paper/2504.03461/) (2025)
- [A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields](https://mlsanity.com/paper/2505.13189/) (2025)
- [A Malliavin calculus approach to score functions in diffusion generative models](https://mlsanity.com/paper/2507.05550/) (2025)
- [Divergence-kernel method for linear responses of densities and generative models](https://mlsanity.com/paper/2509.03992/) (2025)
- [SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation](https://mlsanity.com/paper/2603.22468/) (2026)
- [Conditioning Degenerate Diffusion Models](https://mlsanity.com/paper/2609.04090/) (2026)

## Abstract (arXiv)

We introduce a new framework based on Malliavin calculus to derive exact analytical expressions for the score function nabla log pt(x), i.e., the gradient of the log-density associated with the solution to stochastic differential equations (SDEs). Our approach combines classical integration-by-parts techniques with modern stochastic analysis tools, such as Bismut's formula and Malliavin calculus, and it works for both linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint, the Malliavin divergence (Skorokhod integral), and diffusion generative models, thereby providing a systematic method for computing nabla log pt(x). In the linear case, we present a detailed analysis showing that our formula coincides with the analytical score function derived from the solution of the Fokker--Planck equation. For nonlinear SDEs with state-independent diffusion coefficients, we derive a closed-form expression for nabla log pt(x). We evaluate the proposed framework across multiple generative tasks and find that its performance is comparable to state-of-the-art methods. These results can be generalised to broader classes of SDEs, paving the way for new score-based diffusion generative models.