Carne, T. K. Brownian motion and stereographic projection. (English) Zbl 0566.60080 Ann. Inst. Henri Poincaré, Probab. Stat. 21, 187-196 (1985). 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. Cited in 2 Documents MSC: 60J65 Brownian motion 58J65 Diffusion processes and stochastic analysis on manifolds 53A30 Conformal differential geometry (MSC2010) Keywords:conformal transformations; Stereographic projection PDFBibTeX XMLCite \textit{T. K. Carne}, Ann. Inst. Henri Poincaré, Probab. Stat. 21, 187--196 (1985; Zbl 0566.60080) Full Text: Numdam EuDML