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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.

MSC:

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