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

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.

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.

@article{AIF_1979__29_1_207_0,
     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},
     mrnumber = {81b:30088},
     zbl = {0386.30029},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.735/}
}
TY  - JOUR
AU  - Bernard, Alain
AU  - Campbell, Eddy A.
AU  - Davie, A. M.
TI  - Brownian motion and generalized analytic and inner functions
JO  - Annales de l'Institut Fourier
PY  - 1979
SP  - 207
EP  - 228
VL  - 29
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.735/
DO  - 10.5802/aif.735
LA  - en
ID  - AIF_1979__29_1_207_0
ER  - 
%0 Journal Article
%A Bernard, Alain
%A Campbell, Eddy A.
%A Davie, A. M.
%T Brownian motion and generalized analytic and inner functions
%J Annales de l'Institut Fourier
%D 1979
%P 207-228
%V 29
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.735/
%R 10.5802/aif.735
%G en
%F AIF_1979__29_1_207_0
Bernard, Alain; Campbell, Eddy A.; Davie, A. M. Brownian motion and generalized analytic and inner functions. Annales de l'Institut Fourier, Volume 29 (1979) no. 1, pp. 207-228. doi : 10.5802/aif.735. http://www.numdam.org/articles/10.5802/aif.735/

[1] L.V. Ahlfors, Lectures on Quasi-Conformal Mappings, Van Norstrand, 1966. | MR | Zbl

[2] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Inventiones Math., 7 (1969), 243-268. | EuDML | MR | Zbl

[3] J. Dieudonne, Eléments d'Analyse, Gauthiers-Villars, 1971, Vol. 4. (English translation : Treatise on Analysis, Academic Press, 1974). | Zbl

[4] J. Eells, Singularities of Smooth Maps, Nelson, 1967. | MR | Zbl

[5] L.P. Eisenhart, Riemannian Geometry, Princeton, 1949. | MR | Zbl

[6] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Preprint, Copenhagen University, 1976.

[7] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Nauka Press, Moscow 1964, English translation Academic Press, 1968. | Zbl

[8] N.S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, 1972. | MR | Zbl

[9] H.P. Mckean, Stochastic Integrals, Academic Press, 1969. | MR | Zbl

[10] R. Narasimhan, Introduction to the Theory of analytic Spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, 1966. | MR | Zbl

[11] M.H.A. Newman, Topology of Plane Sets of Points, Cambridge University Press, 2nd. Edition, 1952. | Zbl

[12] I.G. Petrovsky, Lectures on Partial Differential Equations, Interscience, 1954. | MR | Zbl

Cited by Sources: