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We know for planar Brownian motion, that conformal maps composed with Brownian motion are also Brownian motion (preserve distribution).

Does it follow for higher dimensions?

I think it follows for even dimensions because by looking at pair components, the distribution will be preserved and so the joint distribution will be too.

Thanks

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3 Answers 3

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The natural setup for talking about conformal properties of the Brownian motion is that of Riemannian manifolds (by considering the generating operator of the Brownian motion as the Laplace-Beltrami operator of the corresponding Riemannian metric). Now, the Laplacian is conformal in dimension 2 only (in higher dimensions a conformal change of metric gives rise to an additional vector field - this formula is given, for instance, in http://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry#Under_a_conformal_change, although in a somewhat cumbersome form). It means that it is in dimension 2 only that a conformal rescaling of a Riemannian metric amounts to a time rescaling along the sample paths.

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Conformal maps do not preserve the distribution of Brownian motion in the plane. They change it but only by a time reparametrization.

In dimension 3 or more inversion with respect to a sphere produces a process which converges almost surely to the center of the sphere. So it is definitely not a time-change of Brownian motion.

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It is not true. For example, in dimension higher than 2 the BM path tends to infinity, whereas its image under inversion tends to the center of inversion.

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