Brownian motion and generalized analytic and inner functions

Bernard, Alain; Campbell, Eddy A.; Davie, A. M.

Annales de l'Institut Fourier, Tome 29 (1979) no. 1, pp. 207-228.

Soit $f$ une application d’un ouvert ${\mathbf{R}}^p$ dans ${\mathbf{R}}^q$, avec $p>q$. Dire que $f$ conserve le mouvement brownien, à changement de temps aléatoire près, signifie que $f$ est harmonique et que son application linéaire tangente est en chaque point une co-isométrie. Dans le cas $p=2$, $q=2$, ces conditions indiquent que $f$ correspond à une fonction analytique d’une variable complexe. Nous étudions, essentiellement, les cas $p=3$, $q=2$ où nous montrons en particulier qu’une telle application ne peut être “intérieure” sans être triviale. Un résultat analogue pour $p=4$, $q=2$ permettrait de résoudre une conjecture classique sur les fonctions analytiques de deux variables.

Let $f$ be a mapping from an open set in ${\mathbf{R}}^p$ into ${\mathbf{R}}^q$, 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 a classical conjecture on analytic functions of two complex variables.

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