Imagine I have a diffusion process described by the equation $x_1 = f(x_0) + a \cdot \epsilon$, where $\epsilon$ is a random Gaussian number. The way this process evolves from $x_0$ to $x_1$ follows a certain conditional probability called $q(x_1 | x_0)$.
Now, the question is whether I can find a kind of "reverse" diffusion process that goes from $x_1$ back to $x_0$. This reverse process might be represented as $x_0 = g(x_1) + b \cdot \eta$, with $\eta$ being another Gaussian random number. The interesting part is that I want the conditional probability for this reverse process, denoted as $p(x_0 | x_1)$, to be exactly the same as the $q(x_1 | x_0)$ from the original process.