# 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

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topBernard, 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 $p>q$. 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 $p>q$. 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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- [10] R. NARASIMHAN, Introduction to the Theory of analytic Spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, 1966. Zbl0168.06003MR36 #428
- [11] M.H.A. NEWMAN, Topology of Plane Sets of Points, Cambridge University Press, 2nd. Edition, 1952.
- [12] I.G. PETROVSKY, Lectures on Partial Differential Equations, Interscience, 1954. Zbl0059.08402MR16,478f

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