从采样角度系统讲解扩散模型数学原理，适合入门学习。
A Mathematical Introduction to Diffusion Models
- 以采样视角构建扩散模型的数学框架
- 完整推导核心定义与恒等式，辅以简化假设下的估计
- 适合有概率基础的研究生入门，内容层层递进
这些笔记从采样视角出发，对扩散模型进行证明导向的介绍，贯穿从经典采样动力学到现代扩散采样器、误差分析及推理时控制的完整脉络。内容分层展开：核心定义与恒等式完整证明，代表性估计在简化假设下推导，研究级定理仅给出证明路线图。面向具备概率背景但无随机微分方程、随机数值方法或扩散模型先验知识的初阶研究生。
原文摘要 · Abstract (English)
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.
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