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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.


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up vote 3 down vote accepted

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, 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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