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Brownian motion and generalized analytic and inner functions

Alain Bernard; Eddy A. Campbell; A. 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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