Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a random element $x_1$ of $\Omega$ and then assign a measure $s$ on $\Omega$ conditional on $x_1$ (typically $s$ is just a gaussian distribution centered on $x_1$, where the distance is just the distance in $\mathbb R^n$) and then choose high probability point locally and do it again.

It seems to me that this process is subtly decomposing $\Omega$ into a bunch of local spaces which look like $R^m \subset R^n$ with a continuous group action on them, like a lie group with a Haar measure (that happens to be equivalent to $s$) and you're moving along the geodesic lines of the lie group.

I know this is a super vague question, because this is definitely just an analogy, but I was wondering if there is work on something like this? The case I can imagine being workable is $\Omega$ being a discrete space because then it would locally look like lattice and then this sampling procedure might look like optimization on an orbifold. I have no idea for the continuous case, however. Is there any work that sort of looks at this? Or a general area I should look into?