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This work combines, unifies and generalises two of my favourite papers - Ma et al - Complete recipe for autonomous SDEs arxiv.org/abs/1506.04696 - Karras et al - Elucidating the Design Space of Diffusion arxiv.org/abs/2206.00364

Here is the decomposition: I show that the scalar ϕ is unique(!) but you can choose and Q or D. In diffusion the ϕ terms represent the "probability flow ODE" but there are actually many ODEs which satisfy p(x,t) depending on your choice of Q