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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.

Denis
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    You take a standard Brownian motion in the tangent plane and project it using the exponential map. The geodesic properties of that map lead to the involvement of the Dirac/Laplace operator if you express it directly in points of the manifold. // The drift term is a correction that pushes the tangential motion of the $dB$ terms back onto the manifold. – Lutz Lehmann Apr 22 '20 at 12:34
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    See https://math.stackexchange.com/a/3726179/227280 – S.Surace Jun 19 '20 at 11:36

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