Brownian motion and stereographic projection. (English) Zbl 0566.60080

Stereographic projection maps \(R^ N\) to \(S^ N\) conformally. For \(N=2\) this map transforms Brownian paths on \(R^ 2\) into Brownian paths on the sphere \(S^ 2\). The paper shows that, for N exceeding 2, stereographic projection transforms Brownian paths on \(R^ N\) into the paths of Brownian motion on \(S^ N\) which is conditioned to be at the pole of the projection at a negative exponential time. The relation of this property to the conformality of the map is also described.


60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
53A30 Conformal differential geometry (MSC2010)
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