How can we find the SDE of the standard Brownian motion on $S^1$ and $S^2$? In the the following link Brownian motion on $S^1$ , the answer to the question was started with the solution of the SDE, however
1) how one can formally(and informally) obtain the SDE of the two processes?
In the work
The random motion on the sphere generated by the Laplace-Beltrami operator
the SDEs are given for $S^1$ and $S^2$, but there is no explanation for that.
2) Why do we have a drift term in the SDE on $S^2$ (I'm interested in the standard BM)
I would appreciate your answer.