Both panels transport particles from noise (t=1) toward data (t=0) using the same velocity field.
Left (ODE): pure transport, no randomness — density obeys the continuity equation.
Right (SDE): same transport plus Gaussian jitter at each step — density obeys the Fokker–Planck equation.
Watch how the noise term blurs the particle cloud. Press Play.
noise g = 0.8
t = 1.00
ODE (continuity equation)
∂pₜ/∂t = −∇·(pₜ v)
SDE (Fokker–Planck)
∂pₜ/∂t = −∇·(pₜ f) + ½g²Δpₜ
Both start from identical particles. The ODE cloud stays tight — density only flows.
The SDE cloud spreads — the extra ½g²Δp term pushes mass from dense regions into sparse ones.
This is the difference between the continuity equation (pure transport) and the
Fokker–Planck equation (transport + diffusion).