Fokker-Planck Equation and Diffusion Models

Questions that we're going to answer in this document:

1. Why FP equation is necessary to show that stochastic equation used in score-based models preserve the intermediate distribution?
2. Write Fokker-Planck equation. Understand the physical intuition behind it.
3. Derive FP equation, and reverse SDE using FP equation.
4. Derive PF-ODE from the SDE and FP equation.

For the drift function $f:{\mathbb{R}}^d\times \mathbb{R}\to {\mathbb{R}}^d$ and diffusion function $g:{\mathbb{R}}^d\times \mathbb{R}\to {\mathbb{R}}^{d\times d}$ SDE,

$d{X}_t=f(X_t,t)dt+g(X_t,t)d{W}_t$