Chapter 21: Fokker-Planck and Distribution Dynamics. In the chapter mind map this icon labels Paths vs Densities: Three Levels of Description. The discussion below is excerpted and lightly edited from § Three Levels of Description in Mathematics for AI and Machine Learning.

We now have a complete three-level description of stochastic processes:

1. Path level (SDE): Individual random trajectories $\mathbf{X}_t(\omega)$ 2. Path level (ODE): Deterministic trajectories that produce the same marginals 3. Distribution level (PDE): Evolution of the probability density $p_t(\mathbf{x})$

The Fokker-Planck equation provides the highest-level description, directly describing how probability mass flows and spreads, without reference to individual paths. This perspective is particularly powerful for understanding the long-term behavior and convergence properties of diffusion processes.

the book figure visualizes the complete three-level hierarchy of diffusion process descriptions. The top panel shows Level 1: Stochastic Paths (SDE)—individual random trajectories that are noisy and cross each other frequently, representing the physical reality of diffusion. The middle panel shows Level 2: Deterministic Probability Flow (ODE)—smooth, non-crossing paths that define the "skeleton" of the distribution, producing the same marginals as the SDE. The bottom panel shows Level 3: Distributional Evolution (Fokker-Planck PDE)—a heatmap of the probability density $p_t(x)$ evolving over time. Both the SDE and ODE paths above generate samples that follow this exact density evolution. This visualization makes concrete the hierarchy: from random individual paths (microscopic) to deterministic distribution evolution (macroscopic), with the Fokker-Planck equation providing the highest-level mathematical description that unifies all three perspectives.