# Brownian motion and generalized analytic and inner functions

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

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## Abstract

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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.

## 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
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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

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