## 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)

### Keywords:

conformal transformations; Stereographic projection
Full Text: