Clean flow SDE
\[
\begin{cases}
\mathrm{d} \hat{\boldsymbol{x}}^{\text{c}}_{\pm} = \big(\mathrm{d} \big(\frac{\sigma_t}{\alpha_t}\big) \mp \frac{\sigma_t}{\alpha_t} \beta_t \mathrm{d} t\big) \cdot \big(\boldsymbol{\epsilon}_\phi(\alpha_t \hat{\boldsymbol{x}}^{\text{c}}_{\pm} + \sigma_t \tilde{\boldsymbol{\epsilon}}_{\pm}, t, y) - \tilde{\boldsymbol{\epsilon}}_{\pm}\big),\\
\mathrm{d} \tilde{\boldsymbol{\epsilon}}_{\pm} = \mp \tilde{\boldsymbol{\epsilon}}_{\pm} \beta_t \mathrm{d} t + \sqrt{2 \beta_t} \mathrm{d} \boldsymbol{w}_t.
\end{cases}
\]
is equivalent to the following diffusion SDE in EDM [1]:
\[
\begin{align}
\mathrm{d} \big( \frac{\boldsymbol{x}_{\pm}}{\alpha_t} \big)
=& - \sigma_t \nabla_{\boldsymbol{x}_{\pm}} \log p_t(\boldsymbol{x}_{\pm}) \mathrm{d} \big( \frac{\sigma_t}{\alpha_t} \big) \pm \beta_t \big( \frac{\sigma_t}{\alpha_t} \big) \sigma_t \nabla_{\boldsymbol{x}_{\pm}} \log p_t(\boldsymbol{x}_{\pm}) \mathrm{d} t + \sqrt{2 \beta_t} \big( \frac{\sigma_t}{\alpha_t} \big) \mathrm{d} \boldsymbol{w}_t\\
&= \big( \mathrm{d} \big( \frac{\sigma_t}{\alpha_t} \big) \mp \frac{\sigma_t}{\alpha_t} \beta_t \mathrm{d} t \big) \cdot \underbrace{\boldsymbol{\epsilon}_\phi(\boldsymbol{x}_{\pm}, t, y)}_{-\sigma_t \nabla_{\boldsymbol{x}} \log p_t (\boldsymbol{x})} + \sqrt{2 \beta_t} \big( \frac{\sigma_t}{\alpha_t} \big) \mathrm{d} \boldsymbol{w}_t,\label{app:eq:diffusion-sde}
\end{align}
\]
but with \( \hat{\boldsymbol{x}}^{\text{c}}_{\pm} \) being clean images \( \forall t \in [0, T] \) in the whole diffusion process.