A Complete Decomposition of Stochastic Differential Equations

2026/01/12 by Samuel Duffield · 1 voice
Computer Science · Mathematics · #cs.LG #math.PR #math.ST

paper · pdf · doi:10.48550/arxiv.2601.07834

Abstract

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.

Citations

• Scalable Bayesian Learning with posteriors
• Elucidating the Design Space of Diffusion-Based Generative Models
• A Variational Perspective on Diffusion-Based Generative Models and Score Matching
• Time reversal of diffusion processes under a finite entropy condition
• Score-Based Generative Modeling through Stochastic Differential Equations
• Denoising Diffusion Probabilistic Models
• On the geometry of Stein variational gradient descent
• A Conceptual Introduction to Hamiltonian Monte Carlo
• Ensemble preconditioning for Markov chain Monte Carlo simulation
• A Complete Recipe for Stochastic Gradient MCMC
• The Pricing of Options and Corporate Liabilities
• Constructive martingale representation in functional Itô calculus: a local martingale extension

Discussions

• Paper for full details. The proofs draw on ideas from vector calculus and Fourier analysis which was really fun to work through arxiv.org/abs/2601.07834 [bsky, 0 points, 0 comments]

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