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Brownian motion and generalized analytic and inner functions

Alain BernardEddy A. CampbellA. M. Davie — 1979

Annales de l'institut Fourier

Let f be a mapping from an open set in R p into R q , with p > q . To say that f preserves Brownian motion, up to a random change of clock, means that f is harmonic and that its tangent linear mapping in proportional to a co-isometry. In the case p = 2 , q = 2 , such conditions signify that f corresponds to an analytic function of one complex variable. We study, essentially that case p = 3 , q = 2 , in which we prove in particular that such a mapping cannot be “inner” if it is not trivial. A similar result for p = 4 , q = 2 would solve...

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