Brownian motion and generalized analytic and inner functions
Bernard, Alain ; Campbell, Eddy A. ; Davie, A. M.
Annales de l'Institut Fourier, Tome 29 (1979), p. 207-228 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

@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},
     volume = {29},
     year = {1979},
     pages = {207-228},
     doi = {10.5802/aif.735},
     mrnumber = {81b:30088},
     zbl = {0386.30029},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 29 (1979) pp. 207-228. doi : 10.5802/aif.735. http://gdmltest.u-ga.fr/item/AIF_1979__29_1_207_0/

[1] L.V. Ahlfors, Lectures on Quasi-Conformal Mappings, Van Norstrand, 1966. | MR 34 #336 | Zbl 0138.06002

[2] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Inventiones Math., 7 (1969), 243-268. | MR 40 #3445 | Zbl 0183.25901

[3] J. Dieudonne, Eléments d'Analyse, Gauthiers-Villars, 1971, Vol. 4. (English translation : Treatise on Analysis, Academic Press, 1974). | Zbl 0217.00101

[4] J. Eells, Singularities of Smooth Maps, Nelson, 1967. | MR 38 #6612 | Zbl 0167.19903

[5] L.P. Eisenhart, Riemannian Geometry, Princeton, 1949. | MR 11,687g | Zbl 0041.29403

[6] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Preprint, Copenhagen University, 1976. | Zbl 0339.53026

[7] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Nauka Press, Moscow 1964, English translation Academic Press, 1968. | Zbl 0164.13002

[8] N.S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, 1972. | MR 50 #2520 | Zbl 0253.31001

[9] H.P. Mckean, Stochastic Integrals, Academic Press, 1969. | MR 40 #947 | Zbl 0191.46603

[10] R. Narasimhan, Introduction to the Theory of analytic Spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, 1966. | MR 36 #428 | Zbl 0168.06003

[11] M.H.A. Newman, Topology of Plane Sets of Points, Cambridge University Press, 2nd. Edition, 1952. | Zbl 0123.39301

[12] I.G. Petrovsky, Lectures on Partial Differential Equations, Interscience, 1954. | MR 16,478f | Zbl 0059.08402