Theory of Probability and Mathematical Statistics
On spectral theory of random fields in the ball
Nikolai Leonenko, Anatoliy Malyarenko and Andriy Olenko
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Abstract: The paper investigates random fields in the ball. It studies three types of such fields: restrictions of scalar random fields in the ball to the sphere, spin, and vector random fields. The review of the existing results and new spectral theory for each of these classes of random fields are given. Examples of applications to classical and new models of these three types are presented. In particular, the Matérn model is used for illustrative examples. The derived spectral representations can be utilised to further study theoretical properties of such fields and to simulate their realisations. The obtained results can also find various applications for modelling and investigating ball data in cosmology, geosciences and embryology.
Keywords: Random fields, spectral theory, spin, isotropic, random fields in the ball, spherical random fields, Matérn covariance
Bibliography: M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D. C., 1964. MR 0167642
V. V. Anh, A. Olenko, and Y. G. Wang, Fractional stochastic partial differential equation for random tangent fields on the sphere, Theory Probab. Math. Statist. (2021), no. 104, 3–22. MR 4421350
P. Baldi and M. Rossi, Representation of Gaussian isotropic spin random fields, Stochastic Process. Appl. 124 (2014), no. 5, 1910–1941. MR 3170229
Ph. Broadbridge, A. D. Kolesnik, N. Leonenko, and A. Olenko, Random spherical hyperbolic diffusion, J. Stat. Phys. 177 (2019), no. 5, 889–916. MR 4031900
Ph. Broadbridge, A. D. Kolesnik, N. Leonenko, A. Olenko, and D. Omari, Spherically restricted random hyperbolic diffusion, Entropy 22 (2020), no. 2, Paper No. 217, 31. MR 4144958
T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. MR 1410059
O. Christensen, An introduction to frames and Riesz bases, second ed., Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2016. MR 3495345
R. Durrer, The cosmic microwave background, second ed., Cambridge University Press, 2020.
A. Erdélyi, W. Magnus, F. Oberhettinger, and Fr. G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. MR 698779
I. M. Gel′fand and Z. Ya. Šapiro, Representations of the group of rotations in three-dimensional space and their applications, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1(47), 3–117. MR 0047664
D. Geller and D. Marinucci, Spin wavelets on the sphere, J. Fourier Anal. Appl. 16 (2010), no. 6, 840–884. MR 2737761
M. Kamionkowski, A. Kosowsky, and A. Stebbins, Statistics of cosmic microwave background polarization, Phys. Rev. D 55 (1997), 7368–7388.
A. Lang and Chr. Schwab, Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab. 25 (2015), no. 6, 3047–3094. MR 3404631
H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
B. Leistedt, J. D. McEwen, M. Büttner, and H. V. Peiris, Wavelet reconstruction of E and B modes for CMB polarization and cosmic shear analyses, Mon. Not. R. Astron. Soc. 466 (2016), no. 3, 3728–3740.
B. Leistedt, J. D. McEwen, Th. D. Kitching, and H. V. Peiris, 3D weak lensing with spin wavelets on the ball, Phys. Rev. D 92 (2015), 123010.
N. N. Leonenko and L. M. Sakhno, On spectral representations of tensor random fields on the sphere, Stoch. Anal. Appl. 30 (2012), no. 1, 44–66. MR 2870527
H. Luschgy and G. Pagès, Expansions for Gaussian processes and Parseval frames, Electron. J. Probab. 14 (2009), no. 42, 1198–1221. MR 2511282
A. A. Malyarenko, Invariant random fields in vector bundles and application to cosmology, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 4, 1068–1095. MR 2884225
—, Invariant random fields on spaces with a group action, Springer, Heidelberg, 2013. MR 2977490
—, Spectral expansions of cosmological fields, J. Stat. Sci. Appl. 3 (2015), no. 11-12, 175–193.
—, Spectral expansions of random sections of homogeneous vector bundles, Theor. Probability and Math. Statist. (2018), no. 97, 151–165. MR 3746005
D. Marinucci and G. Peccati, Random fields on the sphere. Representation, limit theorems and cosmological applications, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011. MR 2840154
—, Mean-square continuity on homogeneous spaces of compact groups, Electron. Commun. Probab. 18 (2013), 1–10. MR 3064996
R. J. Mathar, Zernike basis to Cartesian transformations, Serb. Astron. J. 179 (2009), 107–120.
E. T. Newman and R. Penrose, Note on the Bondi–Metzner–Sachs group, J. Mathematical Phys. 7 (1966), 863–870. MR 194172
A. M. Obukhov, Statistically homogeneous random fields on a sphere, Uspehi Mat. Nauk 2 (1947), no. 2, 196–198.
Vl. Operstein, Full Müntz theorem in
, J. Approx. Theory 85 (1996), no. 2, 233–235. MR 1385817
Th. W. Pike, Modelling eggshell maculation, Avian Biology Research 8 (2015), no. 4, 237–243.
E. Porcu, M. Bevilacqua, and M. G. Genton, Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere, J. Amer. Statist. Assoc. 111 (2016), no. 514, 888–898. MR 3538713
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2. Special functions, second ed., Gordon & Breach Science Publishers, New York, 1988. MR 950173
K. S. Thorne, Multipole expansions of gravitational radiation, Rev. Modern Phys. 52 (1980), no. 2, part 1, 299–339. MR 569166
A. Trautman, Connections and the Dirac operator on spinor bundles, J. Geom. Phys. 58 (2008), no. 2, 238–252. MR 2384313
M. Volker and K. Seibert, A mathematical view on spin-weighted spherical harmonics and their applications in geodesy, Handbuch der Geodäsie: 6 Bände (Willi Freeden and Reiner Rummel, eds.), Springer, Berlin, Heidelberg, 2019, pp. 1–113.
N. R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
St. Weinberg, Cosmology, Oxford University Press, Oxford, 2008. MR 2410479
M. Ĭ. Yadrenko, Isotropic random fields of Markov type in Euclidean space, Dopovidi Akad. Nauk Ukraïn. RSR 1963 (1963), 304–306. MR 0164376
—, Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983. MR 697386
M. Zaldarriaga and U. Seljak, All-sky analysis of polarization in the microwave background, Phys. Rev. D 55 (1997), 1830–1840.
Fr. von Zernike, Beugungstheorie des Schneidenverfahrens und einer verbesserten Form, der Phasenkontrastmethode, Physica 1 (1934), no. 7, 689–704.
