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Stationary Gaussian random fields on hyperbolic spacesand on Euclidean spheres∗∗∗

Published online by Cambridge University Press:  03 July 2012

S. Cohen  and
M. A. Lifshits
S. Cohen
Affiliation:
Universitéde Toulouse, Université Paul Sabatier, Institut de Mathématiques de Toulouse, 31062 Toulouse, France. Serge.Cohen@math.univ-toulouse.fr
M. A. Lifshits
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl., 2, 198504, Stary Peterhof, Russia; lifts@mail.rcom.ru
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Abstract

We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

Keywords

Hyperbolic spaceRandom fieldsLévy’s Brownian field.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 165 - 221
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

J.W. Anderson, Hyperbolic Geometry, 2nd edition. Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London (2005).
Askey, R. and Bingham, N.H., Gaussian processes on compact symmetric spaces. J. Probab. Theory Relat. Fields 37 (1976) 127–143. Google Scholar
Barsky, S., Surface texture using photometric stereo data : classification and direction of illumination detection. J. Math. Imaging Vis. 29 (2007) 185–204. Google Scholar
Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J.-L., Lois stables et espaces Lp. Ann. Inst. Henri Poincaré, Ser. B. 2 (1965/66) 231–259. Google Scholar
J.W. Cannon, W.J. Floyd, R. Kenyon and W.R. Parry, Hyperbolic geometry, in Flavors of Geometry, edited by S. Levy. Cambridge University Press, Cambridge. Math. Sci. Res. Inst. Publ. 31 (1997) 59–115.
Chentsov, N.N., Lévy Brownian Motion for several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265–266. Google Scholar
Chentsov, N.N. and Morozova, E.A., P. Lévy’s random fields. Theory Probab. Appl. 12 (1967) 153–156. Google Scholar
Clerc, M. and Mallat, S., Estimating deformations of stationary processes. Ann. Stat. 31 (2003) 1772–1821. Google Scholar
J.L. Clerc, J. Faraut, M. Rais, P. Eymard and R. Takahashi, Analyse Harmonique. Les Cours du CIMPA (1980).
Dunau, J.-L. and Senateur, H., Characterization of the type of some generalizations of the Cauchy distribution, in Probability measures on Groups IX. Oberwolfach (1988). Lect. Notes Math. 1379 (1989) 64–74. Google Scholar
Faraut, J. and Harzallah, K., Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24 (1974) 171–217. Google Scholar
Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. Henri Poincaré Sect. B (N.S.) 3 (1967) 121–226. Google Scholar
Garding, J., Shape from texture and contour by weak isotropy. Artif. Intell. 64 (1993) 243–297. Google Scholar
R. Godement, Introductions aux travaux de A. Selberg, Séminaire Bourbaki (1957) 95–110.
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, VI edition. Academic Press, New York (2000).
S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, 2nd edition. Academic Press 80 (1978).
S. Helgason, Groups and Geometric Analysis, edited by American Mathematical Society, Providence, RI. Mathematical Surveys and Monographs 83 (2000). Integral geometry, invariant differential operators, and spherical functions. Corrected reprint of the 1984 original.
Istas, J., Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254–262 (electronic). Google Scholar
Istas, J., On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242–251 (electronic). Google Scholar
J. Istas, Manifold indexed fractional fields. Preprint (2009).
N.L. Johnson and S. Kotz, Distributions in statistics : continuous multivariate distributions. Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York (1972).
P. Lévy, Processus Stochastiques et Mouvement Brownien, 2éme édition, edited by J. Gabay (1965).
E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 2nd edition. American Mathematical Society, Providence, RI 14 (2001).
Lifshits, M.A., On the representation of Lévy fields by indicators. Theory Probab. Appl. 24 (1980) 629–633. Google Scholar
M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers (1995).
McKean, H.P., Brownian Motion with a several-dimensional time. Theory Probab. Appl. 8 (1963) 335–354. Google Scholar
Molchan, G.M., On some problems concerning Brownian motion in Lévy’s sense. Theory Probab. Appl. 12 (1967) 682–690. Google Scholar
G.M. Molchan, On homogenious random fields on symmetric spaces of rank 1(Russian). Teor. Veroyatnost. i Mat. Statist. (1979) 123–147. Translated in : Theor. Probab. Math. Statist. (1980) 143–168.
G.M. Molchan, Multiparametric Brownian motion on symmetric spaces. VNU Sci. Press, Utrecht (1987). Prob. Theory and Math. Stat. II. Vilnius (1985) 275–286.
G.M. Molchan, Multiparameter Brownian motion (Russian). Teor. Veroyatnost. i Mat. Statist. (1987) 88–101. Translated in : Theor. Probab. Math. Statist. (1988) 97–110.
G.M. Molchan, Private communication (2009).
Robertson, A.G., Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163–172. Google Scholar
W. Rudin, Fourier Analysis on Groups. Wiley Classics Library, John Wiley & Sons Inc., New York (1990). Reprint of the 1962 original, A Wiley-Interscience Publication.
Santaló, L.A., Integral geometry on surfaces of constant negative curvature. Duke Math. J. 10 (1943) 687–709. Google Scholar
R. Stanton and P.Thomas, Expansions of spherical functions on non-compact spaces, Acta Math. 40 (1978) 251–276.
D.W. Stroock, The Ornstein-Uhlenbeck process in a Riemanian manifold, in Proc. of ICCM’98 (Beijing, 1998), First International congress of Chinese Mathematicians. AMS (2001) 11–23.
Takenaka, S., Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1–12. Google Scholar
Takenaka, S., Kubo, I. and Urakawa, H., Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131–140. Google Scholar
N.A. Volodin, Some classes of spherically symmetric distributions. Stability problems for stochastic models (Russian) Sukhumi (1987), Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow (1988), Translated in J. Soviet Math. 57 (1991) 3189–3192, 4–8.
A.M. Yaglom, An Introduction to the Theory of Stationary Random Functions. Revised English edition, Prentice-Hall Inc., Englewood Cliffs, N.J. (1962)