Density Evolution

Chapter 19: Stochastic Differential Equations. In the chapter mind map this icon labels Probability Evolution: Fokker-Planck Preview. The discussion below is excerpted and lightly edited from § Probability Evolution Viewpoint in Mathematics for AI and Machine Learning. Related material also appears in Chapter 21 (Paths vs Densities: Three Levels of Description), Chapter 21 (Determinism of Prob Distributions).

While individual trajectories $\mathbf{X}_t$ are stochastic, an initial distribution $\mathbf{X}_0 \sim p_0(\mathbf{x})$ produces an ensemble density $p_t(\mathbf{x})$ that evolves completely deterministically.

By conservation of probability mass, the local density change equals the negative divergence of the probability current $\mathbf{J}_t(\mathbf{x})$:

When the diffusion does not depend on the state, $\mathbf{G}(\mathbf{X}_t, t) = \mathbf{G}(t)$, the flux balances deterministic drift transport $\mathbf{f}(\mathbf{x}, t) p_t(\mathbf{x})$ and diffusive spreading $-\frac{1}{2}\mathbf{a}(t)\nabla_{\mathbf{x}} p_t(\mathbf{x})$, where $\mathbf{a}(t) = \mathbf{G}(t)\mathbf{G}(t)^\top$. Substituting $\mathbf{J}_t(\mathbf{x})$ yields the Fokker–Planck equation (developed, for state-dependent $\mathbf{G}$ as well, in the matrix chapter):

This macroscopic view shifts the objective from simulating isolated sample paths to controlling the continuous evolution of distributions—the foundation for reversing time in generative modeling.