## Electronic Journal of Probability

### Nonexistence of fractional Brownian fields indexed by cylinders

Nil Venet

#### Abstract

We show in this article that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S} ^{1} \times ]0,\varepsilon [$ endowed with its product distance $d$ for any $\varepsilon >0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels – or valid covariances – on the cylinder.

We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle.

As a consequence of our result, we show that the set of $H$ such that $d^{2H}$ is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces.

These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 75, 26 pp.

Dates
Accepted: 7 December 2018
First available in Project Euclid: 3 July 2019

https://projecteuclid.org/euclid.ejp/1562119475

Digital Object Identifier
doi:10.1214/18-EJP256

Mathematical Reviews number (MathSciNet)
MR3978225

Zentralblatt MATH identifier
07089013

#### Citation

Venet, Nil. Nonexistence of fractional Brownian fields indexed by cylinders. Electron. J. Probab. 24 (2019), paper no. 75, 26 pp. doi:10.1214/18-EJP256. https://projecteuclid.org/euclid.ejp/1562119475

#### References

• [1] C. Berg, J. P. R. Christensen, and P. Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984, Theory of positive definite and related functions.
• [2] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
• [3] S. Cohen and M. A. Lifshits, Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres, ESAIM Probab. Stat. 16 (2012), 165–221.
• [4] M. P. do Carmo, Differential geometry of curves & surfaces, Dover Publications, Inc., Mineola, NY, 2016, Revised & updated second edition of [MR-0394451].
• [5] J. Faraut and K. Harzallah, Distances hilbertiennes invariantes sur un espace homogène, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiv, 171–217.
• [6] A. Feragen, F. Lauze, and S. Hauberg, Geodesic exponential kernels: When curvature and linearity conflict, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.
• [7] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, third ed., Universitext, Springer-Verlag, Berlin, 2004.
• [8] R. Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967), 121–226.
• [9] J. Istas, Spherical and hyperbolic fractional Brownian motion, Electron. Comm. Probab. 10 (2005), 254–262.
• [10] J. Istas, Karhunen-Loève expansion of spherical fractional Brownian motions, Statist. Probab. Lett. 76 (2006), no. 14, 1578–1583.
• [11] J. Istas, Manifold indexed fractional fields, ESAIM Probab. Stat. 16 (2012), 222–276.
• [12] A. Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005.
• [13] P. Lévy, Le mouvement Brownien fonction d’un point de la sphère de Riemann., Rend. Circ. Mat. Palermo, II. Ser. 8 (1960), 297–310 (French).
• [14] P. Lévy, Processus stochastiques et mouvement brownien, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992, Followed by a note by M. Loève, Reprint of the second (1965) edition.
• [15] M. A. Lifshits, Lectures on Gaussian processes, SpringerBriefs in Mathematics, Springer, Heidelberg, 2012.
• [16] M. A. Lifshits, The representation of Lévy fields by indicators, Teor. Veroyatnost. i Primenen. 24 (1979), no. 3, 624–628.
• [17] B. B. Mandelbrot, Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands, Proc. Nat. Acad. Sci. U. S. A. 72 (1975), no. 10, 3825–3828.
• [18] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437.
• [19] G. M. Molchan, Multiparametric Brownian motion on symmetric spaces, Probability theory and mathematical statistics, Vol. II (Vilnius, 1985), VNU Sci. Press, Utrecht, 1987, pp. 275–286.
• [20] E. A. Morozova and N. N. Chentsov, Lévy’s random fields, Teor. Verojatnost. i Primenen 13 (1968), 152–155.
• [21] B. Schölkopf and A. J. Smola, Learning with kernels: support vector machines, regularization, optimization, and beyond, MIT press, 2002.
• [22] S. Takenaka, Representation of Euclidean random field, Nagoya Math. J. 105 (1987), 19–31.
• [23] N. Venet, On the existence of fractional Brownian fields indexed by manifolds with closed geodesics, arXiv:1612.05984