Perturbing data to noise with a continuous-time stochastic process (when the num of noise scales approaches infinity)
Many stochastic processes(diffusion processes in particular) are solutions of stochastic differential equations(SDEs).
In general, SDEs follow this equation
$$ \begin{equation} \mathrm{d}\mathbf{x} = \mathbf{f}(\mathbf{x}, t) \mathrm{d}t + g(t) \mathrm{d} \mathbf{w}, \end{equation} $$
the choice of SDEs is not unique, e.g. $\mathrm{d}\mathbf{x} = e^{t} \mathrm{d} \mathbf{w}$
Yang Song et al. provide 3 choices of SDE: the Variance Exploding SDE (VE SDE), the Variance Preserving SDE (VP SDE), and the sub-VP SDE\