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.
Classification : 60G15, 60G10, 51M10
Mots clés : hyperbolic space, random fields, Lévy's brownian field
@article{PS_2012__16__165_0,
author = {Cohen, S. and Lifshits, M. A.},
title = {Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres},
journal = {ESAIM: Probability and Statistics},
pages = {165--221},
publisher = {EDP-Sciences},
volume = {16},
year = {2012},
doi = {10.1051/ps/2011105},
zbl = {1275.60038},
mrnumber = {2946126},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2011105/}
}
TY - JOUR AU - Cohen, S. AU - Lifshits, M. A. TI - Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres JO - ESAIM: Probability and Statistics PY - 2012 DA - 2012/// SP - 165 EP - 221 VL - 16 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2011105/ UR - https://zbmath.org/?q=an%3A1275.60038 UR - https://www.ams.org/mathscinet-getitem?mr=2946126 UR - https://doi.org/10.1051/ps/2011105 DO - 10.1051/ps/2011105 LA - en ID - PS_2012__16__165_0 ER -
Cohen, S.; Lifshits, M. A. Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 165-221. doi : 10.1051/ps/2011105. http://archive.numdam.org/articles/10.1051/ps/2011105/
[1] , Hyperbolic Geometry, 2nd edition. Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London (2005). | MR 2161463 | Zbl 0934.51012
[2] and , Gaussian processes on compact symmetric spaces. J. Probab. Theory Relat. Fields 37 (1976) 127-143. | MR 423000 | Zbl 0329.60019
[3] , Surface texture using photometric stereo data : classification and direction of illumination detection. J. Math. Imaging Vis. 29 (2007) 185-204. | MR 2385289
[4] , and , Lois stables et espaces Lp. Ann. Inst. Henri Poincaré, Ser. B. 2 (1965/66) 231-259. | Numdam | MR 203757 | Zbl 0139.33501
[5] , , and , Hyperbolic geometry, in Flavors of Geometry, edited by S. Levy. Cambridge University Press, Cambridge. Math. Sci. Res. Inst. Publ. 31 (1997) 59-115. | MR 1491098 | Zbl 0899.51012
[6] , Lévy Brownian Motion for several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265-266.
[7] and , P. Lévy's random fields. Theory Probab. Appl. 12 (1967) 153-156. | Zbl 0196.18702
[8] and , Estimating deformations of stationary processes. Ann. Stat. 31 (2003) 1772-1821. | MR 2036390 | Zbl 1052.62086
[9] , , , and , Analyse Harmonique. Les Cours du CIMPA (1980). | Zbl 0569.43002
[10] and , 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. | MR 1020522 | Zbl 0681.60020
[11] and , Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24 (1974) 171-217. | Numdam | MR 365042 | Zbl 0265.43013
[12] , 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. | Numdam | MR 215331 | Zbl 0157.24902
[13] , Shape from texture and contour by weak isotropy. Artif. Intell. 64 (1993) 243-297. | MR 1259579 | Zbl 0942.68756
[14] , Introductions aux travaux de A. Selberg, Séminaire Bourbaki (1957) 95-110. | Numdam | MR 1610957 | Zbl 0202.40902
[15] and , Table of Integrals, Series, and Products, VI edition. Academic Press, New York (2000). | MR 1398882 | Zbl 0521.33001
[16] , Differential Geometry, Lie Groups and Symmetric Spaces, 2nd edition. Academic Press 80 (1978). | MR 514561 | Zbl 0451.53038
[17] , 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. | MR 1790156 | Zbl 0965.43007
[18] , Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254-262 (electronic). | MR 2198600 | Zbl 1112.60029
[19] , On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242-251 (electronic). | MR 2266715 | Zbl 1110.60032
[20] , Manifold indexed fractional fields. Preprint (2009). | Numdam | MR 2956575 | Zbl 1275.60041
[21] and , Distributions in statistics : continuous multivariate distributions. Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York (1972). | MR 418337 | Zbl 0248.62021
[22] , Processus Stochastiques et Mouvement Brownien, 2éme édition, edited by J. Gabay (1965). | Zbl 0034.22603
[23] and , Analysis, Graduate Studies in Mathematics, 2nd edition. American Mathematical Society, Providence, RI 14 (2001). | MR 1817225 | Zbl 0966.26002
[24] , On the representation of Lévy fields by indicators. Theory Probab. Appl. 24 (1980) 629-633. | Zbl 0446.60038
[25] , Gaussian Random Functions. Kluwer Academic Publishers (1995). | MR 1472736 | Zbl 0832.60002
[26] , Brownian Motion with a several-dimensional time. Theory Probab. Appl. 8 (1963) 335-354. | MR 157407 | Zbl 0124.08702
[27] , On some problems concerning Brownian motion in Lévy's sense. Theory Probab. Appl. 12 (1967) 682-690. | Zbl 0159.46504
[28] , 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. | MR 550252 | Zbl 0485.60054
[29] , Multiparametric Brownian motion on symmetric spaces. VNU Sci. Press, Utrecht (1987). Prob. Theory and Math. Stat. II. Vilnius (1985) 275-286. | MR 901539 | Zbl 0653.58045
[30] , Multiparameter Brownian motion (Russian). Teor. Veroyatnost. i Mat. Statist. (1987) 88-101. Translated in : Theor. Probab. Math. Statist. (1988) 97-110. | MR 913723 | Zbl 0659.60073
[31] , Private communication (2009).
[32] , Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163-172. | MR 1616751 | Zbl 0891.43005
[33] , Fourier Analysis on Groups. Wiley Classics Library, John Wiley & Sons Inc., New York (1990). Reprint of the 1962 original, A Wiley-Interscience Publication. | MR 1038803 | Zbl 0698.43001
[34] , Integral geometry on surfaces of constant negative curvature. Duke Math. J. 10 (1943) 687-709. | MR 9469 | Zbl 0063.06703
[35] and , Expansions of spherical functions on non-compact spaces, Acta Math. 40 (1978) 251-276. | Zbl 0411.43014
[36] , 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. | MR 1830163 | Zbl 1054.60084
[37] , Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1-12. | MR 1126180 | Zbl 0757.60035
[38] , and , Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131-140. | MR 618812 | Zbl 0483.60008
[39] , 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. | MR 1079116 | Zbl 0799.60016
[40] , An Introduction to the Theory of Stationary Random Functions. Revised English edition, Prentice-Hall Inc., Englewood Cliffs, N.J. (1962) | MR 184289 | Zbl 0121.12601
Cité par Sources :
