Deriving Reverse-Time Stochastic Differential Equations (SDEs) (jiha-kim.github.io)

This note gives a self-contained derivation of the reverse-time Itô SDE for a general diffusion X_t driven by dX_t = f(X_t,t) dt + g(X_t,t) dW_t, filling a gap in commonly cited treatments (e.g., Song et al. 2021 referring to Anderson 1982). By rewriting the forward Fokker–Planck equation under the time change s = T − t, postulating a reverse SDE for Y_s = X_{T−s}, and equating the two density evolutions (using integration by parts and test functions), the author isolates the exact reverse drift. Under natural regularity and ellipticity assumptions the reverse drift is b(x,s) = −F(x,s) + ∇[G(x,s)^2] + G(x,s)^2 ∇ log q(x,s), where F(x,s)=f(x,T−s), G(x,s)=g(x,T−s), and q is the reverse-time density. The G^2 ∇ log q term is the familiar score-weighted diffusion term used in score-based generative modeling; the extra ∇G^2 term appears when the diffusion coefficient depends on space. This clarification matters for the AI/ML community because many diffusion-model algorithms assume time-only noise schedules; the derivation shows precisely when that simplification is valid and what correction is needed for space-dependent noise. The approach highlights key technical ingredients (Fokker–Planck, positivity/smoothness of densities, reversed filtration) that guarantee the reverse SDE representation.