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Suppose I have an $n$-dimensional manifold $M$ with a chart $\left(x,U\right)$. Are there any known methods for simulating Brownian motion on $M$ by first simulating a process in $x\left(U\right)\subset\mathbb{R}^n$ and mapping the solution back to $M$ via the map $x^{-1}:x\left(U\right)\longrightarrow U\subset M$? Thanks.

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