reverse-time SDE

The reverse-time SDE is the time-reversed version of a forward diffusion process, established by Anderson (1982). Given a forward Itô SDE $dx=f(x,t)dt+g(t)dw$, its reverse-time counterpart is:

$dx=[f(x,t)-g(t{)}^2{\nabla }_x\log p_t(x)]d\bar{t}+g(t)d\bar{w}$

where $d\bar{t}$ denotes reverse time flow and $\bar{w}$ is a reverse-time Wiener process. The key insight is that the reverse drift depends only on the score function ${\nabla }_x\log p_t(x)$ of the forward process marginals at each time.

This result is the theoretical foundation of score-based generative modeling: by training a neural network to estimate the score, one can simulate the reverse-time SDE to generate samples from the data distribution starting from noise. It also connects to Nelson’s stochastic mechanics: the forward and backward drifts differ by exactly ${\sigma }^2\nabla \log p_t$ (Nelson’s osmotic velocity).

For Gaussian random bridges, the reverse-time SDE simplifies dramatically: the score terms cancel exactly, yielding $d{\bar{\xi }}_t=({\bar{\xi }}_t-x)/td\bar{t}+\sigma d{\bar{W}}_t$ — a pure source-retracting drift with no score dependence (Shelley & Mengütürk 2025).

Key Details

• Anderson (1982) / Haussmann-Pardoux result
• Reverse drift = forward drift $-g^2\nabla \log p_t$
• Requires only score function estimation
• Foundation of all SDE-based generative models
• For Gaussian bridges, score cancels in reverse drift
• Connects to Nelson’s osmotic velocity
• Also used by Schrödinger bridge methods

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concept