Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres

Cohen, S.; Lifshits, M. A.

ESAIM: Probability and Statistics, Tome 16 (2012), pp. 165-221.

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

[1] J.W. Anderson, Hyperbolic Geometry, 2nd edition. Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London (2005). | MR | Zbl

[2] R. Askey and N.H. Bingham, Gaussian processes on compact symmetric spaces. J. Probab. Theory Relat. Fields 37 (1976) 127-143. | MR | Zbl

[3] S. Barsky, Surface texture using photometric stereo data : classification and direction of illumination detection. J. Math. Imaging Vis. 29 (2007) 185-204. | MR

[4] J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine, Lois stables et espaces Lp. Ann. Inst. Henri Poincaré, Ser. B. 2 (1965/66) 231-259. | Numdam | MR | Zbl

[5] 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. | MR | Zbl

[6] N.N. Chentsov, Lévy Brownian Motion for several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265-266.

[7] N.N. Chentsov and E.A. Morozova, P. Lévy's random fields. Theory Probab. Appl. 12 (1967) 153-156. | Zbl

[8] M. Clerc and S. Mallat, Estimating deformations of stationary processes. Ann. Stat. 31 (2003) 1772-1821. | MR | Zbl

[9] J.L. Clerc, J. Faraut, M. Rais, P. Eymard and R. Takahashi, Analyse Harmonique. Les Cours du CIMPA (1980). | Zbl

[10] J.-L. Dunau and H. Senateur, 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 | Zbl

[11] J. Faraut and K. Harzallah, Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24 (1974) 171-217. | Numdam | MR | Zbl

[12] R. Gangolli, 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 | Zbl

[13] J. Garding, Shape from texture and contour by weak isotropy. Artif. Intell. 64 (1993) 243-297. | MR | Zbl

[14] R. Godement, Introductions aux travaux de A. Selberg, Séminaire Bourbaki (1957) 95-110. | Numdam | MR | Zbl

[15] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, VI edition. Academic Press, New York (2000). | MR | Zbl

[16] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, 2nd edition. Academic Press 80 (1978). | MR | Zbl

[17] 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. | MR | Zbl

[18] J. Istas, Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254-262 (electronic). | MR | Zbl

[19] J. Istas, On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242-251 (electronic). | MR | Zbl

[20] J. Istas, Manifold indexed fractional fields. Preprint (2009). | Numdam | MR | Zbl

[21] 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). | MR | Zbl

[22] P. Lévy, Processus Stochastiques et Mouvement Brownien, 2éme édition, edited by J. Gabay (1965). | Zbl

[23] E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 2nd edition. American Mathematical Society, Providence, RI 14 (2001). | MR | Zbl

[24] M.A. Lifshits, On the representation of Lévy fields by indicators. Theory Probab. Appl. 24 (1980) 629-633. | Zbl

[25] M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers (1995). | MR | Zbl

[26] H.P. Mckean, Brownian Motion with a several-dimensional time. Theory Probab. Appl. 8 (1963) 335-354. | MR | Zbl

[27] G.M. Molchan, On some problems concerning Brownian motion in Lévy's sense. Theory Probab. Appl. 12 (1967) 682-690. | Zbl

[28] 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. | MR | Zbl

[29] G.M. Molchan, Multiparametric Brownian motion on symmetric spaces. VNU Sci. Press, Utrecht (1987). Prob. Theory and Math. Stat. II. Vilnius (1985) 275-286. | MR | Zbl

[30] G.M. Molchan, Multiparameter Brownian motion (Russian). Teor. Veroyatnost. i Mat. Statist. (1987) 88-101. Translated in : Theor. Probab. Math. Statist. (1988) 97-110. | MR | Zbl

[31] G.M. Molchan, Private communication (2009).

[32] A.G. Robertson, Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163-172. | MR | Zbl

[33] 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. | MR | Zbl

[34] L.A. Santaló, Integral geometry on surfaces of constant negative curvature. Duke Math. J. 10 (1943) 687-709. | MR | Zbl

[35] R. Stanton and P. Thomas, Expansions of spherical functions on non-compact spaces, Acta Math. 40 (1978) 251-276. | Zbl

[36] 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. | MR | Zbl

[37] S. Takenaka, Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1-12. | MR | Zbl

[38] S. Takenaka, I. Kubo and H. Urakawa, Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131-140. | MR | Zbl

[39] 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. | MR | Zbl

[40] A.M. Yaglom, An Introduction to the Theory of Stationary Random Functions. Revised English edition, Prentice-Hall Inc., Englewood Cliffs, N.J. (1962) | MR | Zbl