Brownian motion and generalized analytic and inner functions

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

Annales de l'institut Fourier (1979)

  • Volume: 29, Issue: 1, page 207-228
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Bernard, Alain, Campbell, Eddy A., and Davie, A. M.. "Brownian motion and generalized analytic and inner functions." Annales de l'institut Fourier 29.1 (1979): 207-228. <http://eudml.org/doc/74397>.

@article{Bernard1979,
abstract = {Let $f$ be a mapping from an open set in $\{\bf R\}^p$ into $\{\bf R\}^q$, with $p&gt;q$. 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, Campbell, Eddy A., Davie, A. M.},
journal = {Annales de l'institut Fourier},
keywords = {GENERALIZED ANALYTIC FUNCTIONS; BROWNIAN MOTIONS; PROBABILISTIC POTENTIAL THEORY},
language = {eng},
number = {1},
pages = {207-228},
publisher = {Association des Annales de l'Institut Fourier},
title = {Brownian motion and generalized analytic and inner functions},
url = {http://eudml.org/doc/74397},
volume = {29},
year = {1979},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 29
IS - 1
SP - 207
EP - 228
AB - Let $f$ be a mapping from an open set in ${\bf R}^p$ into ${\bf R}^q$, with $p&gt;q$. 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.
LA - eng
KW - GENERALIZED ANALYTIC FUNCTIONS; BROWNIAN MOTIONS; PROBABILISTIC POTENTIAL THEORY
UR - http://eudml.org/doc/74397
ER -

References

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  1. [1] L.V. AHLFORS, Lectures on Quasi-Conformal Mappings, Van Norstrand, 1966. Zbl0138.06002MR34 #336
  2. [2] E. BOMBIERI, E. DE GIORGI and E. GIUSTI, Minimal cones and the Bernstein problem, Inventiones Math., 7 (1969), 243-268. Zbl0183.25901MR40 #3445
  3. [3] J. DIEUDONNE, Eléments d'Analyse, Gauthiers-Villars, 1971, Vol. 4. (English translation : Treatise on Analysis, Academic Press, 1974). Zbl0217.00101
  4. [4] J. EELLS, Singularities of Smooth Maps, Nelson, 1967. Zbl0167.19903MR38 #6612
  5. [5] L.P. EISENHART, Riemannian Geometry, Princeton, 1949. Zbl0041.29403MR11,687g
  6. [6] B. FUGLEDE, Harmonic morphisms between Riemannian manifolds, Preprint, Copenhagen University, 1976. 
  7. [7] O.A. LADYZHENSKAYA and N.N. URAL'TSEVA, Linear and Quasilinear Elliptic Equations, Nauka Press, Moscow 1964, English translation Academic Press, 1968. Zbl0164.13002
  8. [8] N.S. LANDKOF, Foundations of Modern Potential Theory, Springer-Verlag, 1972. Zbl0253.31001MR50 #2520
  9. [9] H.P. McKEAN, Stochastic Integrals, Academic Press, 1969. Zbl0191.46603MR40 #947
  10. [10] R. NARASIMHAN, Introduction to the Theory of analytic Spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, 1966. Zbl0168.06003MR36 #428
  11. [11] M.H.A. NEWMAN, Topology of Plane Sets of Points, Cambridge University Press, 2nd. Edition, 1952. 
  12. [12] I.G. PETROVSKY, Lectures on Partial Differential Equations, Interscience, 1954. Zbl0059.08402MR16,478f

Citations in EuDML Documents

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  1. Bernt Oksendal, Daniel W. Stroock, A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations translations and dilatations
  2. G. Letac, La préservation des trajectoires du mouvement brownien et la préservation des lois de Cauchy
  3. M. Yor, À propos de l’inverse du mouvement brownien dans ℝ n ( n ≧ 3 )
  4. Laurent Danielo, Construction de métriques d’Einstein à partir de transformations biconformes
  5. R. W. R. Darling, Martingales in manifolds. Definition, examples and behaviour under maps
  6. Bernt Oksendal, L. Csink, Stochastic harmonic morphisms : functions mapping the paths of one diffusion into the paths of another
  7. Frédérique Duheille, Une preuve probabiliste élémentaire d'un résultat de P. Baird et J. C. Wood
  8. Paul Baird, Harmonic morphisms onto Riemann surfaces and generalized analytic functions
  9. J. M. Coron, F. Helein, Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry
  10. Ye-Lin Ou, Quadratic harmonic morphisms and O-systems

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