A Mathematical Introduction to Diffusion Models

Abstract

These notes give a proof-oriented introduction to diffusion models from the viewpoint of sampling, tracing a single arc from classical sampling dynamics to modern diffusion samplers, their error analysis, and inference-time control. Throughout, the material is layered into core definitions and identities proved in full, representative estimates proved under simplifying assumptions, and research-level theorems stated with a proof roadmap. The intended audience is beginning graduate students with a background in probability but no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models.

Lecture notes for the John Tukey Summer Graduate School on Mathematics of Generative Models at SLMath (June 22nd, 2026 -- July 2nd, 2026)