Mathematical Introduction to Diffusion Models for Graduate Students

Diffusion models, a powerful class of generative models, are gaining widespread attention, but their underlying mathematical principles can be complex. This new set of notes offers a structured, proof-oriented introduction to these models, making them accessible to a specific audience. The material systematically connects classical sampling dynamics to the sophisticated diffusion samplers used today. The introduction covers essential definitions and identities, providing full proofs for core concepts. It also includes representative estimates, proved under simplifying assumptions, and outlines proof roadmaps for more advanced, research-level theorems. This layered approach aims to build a solid foundational understanding. The target audience for these notes is beginning graduate students who possess a strong background in probability theory. Crucially, it assumes no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models themselves, making it an ideal starting point for those looking to delve into the mathematical underpinnings of this rapidly evolving AI field.