Brownian motion and generalized analytic and inner functions
Annales de l'Institut Fourier, Tome 29 (1979) no. 1, pp. 207-228.

Soit f une application d’un ouvert R p dans 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 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 a classical conjecture on analytic functions of two complex variables.

     author = {Bernard, Alain and Campbell, Eddy A. and Davie, A. M.},
     title = {Brownian motion and generalized analytic and inner functions},
     journal = {Annales de l'Institut Fourier},
     pages = {207--228},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {1},
     year = {1979},
     doi = {10.5802/aif.735},
     zbl = {0386.30029},
     mrnumber = {81b:30088},
     language = {en},
     url = {}
Bernard, Alain; Campbell, Eddy A.; Davie, A. M. Brownian motion and generalized analytic and inner functions. Annales de l'Institut Fourier, Tome 29 (1979) no. 1, pp. 207-228. doi : 10.5802/aif.735.

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