Fractional Brownian fields over manifolds
HTML articles powered by AMS MathViewer
- by Zachary A. Gelbaum PDF
- Trans. Amer. Math. Soc. 366 (2014), 4781-4814 Request permission
Abstract:
Extensions of the fractional Brownian fields are constructed over a complete Riemannian manifold. This construction is carried out for the full range of the Hurst parameter $\alpha \in (0,1)$. In particular, we establish existence, distributional scaling (self-similiarity), stationarity of the increments, and almost sure Hölder continuity of sample paths. Stationary counterparts to these fields are also constructed.References
- Robert J. Adler, The geometry of random fields, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1981. MR 611857
- Robert J. Adler and Jonathan E. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516
- A. Benassi, Locally self-similar Gaussian processes, Wavelets and statistics (Villard de Lans, 1994) Lect. Notes Stat., vol. 103, Springer, New York, 1995, pp. 43–54. MR 1364671, DOI 10.1007/978-1-4612-2544-7_{4}
- Albert Benassi, Stéphane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90 (English, with English and French summaries). MR 1462329, DOI 10.4171/RMI/217
- Albert Benassi and Daniel Roux, Elliptic self-similar stochastic processes, Rev. Mat. Iberoamericana 19 (2003), no. 3, 767–796. MR 2053563, DOI 10.4171/RMI/369
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
- Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
- Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
- E. B. Davies, Pointwise bounds on the space and time derivatives of heat kernels, J. Operator Theory 21 (1989), no. 2, 367–378. MR 1023321
- E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR 1103113
- E. B. Davies, The state of the art for heat kernel bounds on negatively curved manifolds, Bull. London Math. Soc. 25 (1993), no. 3, 289–292. MR 1209255, DOI 10.1112/blms/25.3.289
- Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI 10.1512/iumj.1983.32.32046
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Ramesh 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. MR 0215331
- Adriano M. Garsia, Continuity properties of Gaussian processes with multidimensional time parameter, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 369–374. MR 0410880
- Alexander Grigor′yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), no. 1, 33–52. MR 1443330
- Alexander Grigor’yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR 2569498, DOI 10.1090/amsip/047
- P. G. Hjorth, S. L. Kokkendorff, and S. Markvorsen, Hyperbolic spaces are of strictly negative type, Proc. Amer. Math. Soc. 130 (2002), no. 1, 175–181. MR 1855636, DOI 10.1090/S0002-9939-01-06056-7
- Jacques Istas, Spherical and hyperbolic fractional Brownian motion, Electron. Comm. Probab. 10 (2005), 254–262. MR 2198600, DOI 10.1214/ECP.v10-1166
- Jacques Istas, On fractional stable fields indexed by metric spaces, Electron. Comm. Probab. 11 (2006), 242–251. MR 2266715, DOI 10.1214/ECP.v11-1223
- Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
- N. C. Lee, Estimates for heat kernel and Green’s function on certain manifolds with Ricci curvature bounded below, Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991) Int. Press, Cambridge, MA, 1997, pp. 247–258. MR 1482041
- Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1965 (French). Suivi d’une note de M. Loève; Deuxième édition revue et augmentée. MR 0190953
- Peter Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1–21. MR 847950, DOI 10.2307/1971385
- H. P. McKean, An upper bound to the spectrum of $\Delta$ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359–366. MR 266100
- G. M. Molchan, Multiparameter Brownian motion, Teor. Veroyatnost. i Mat. Statist. 36 (1987), 88–101, 141 (Russian); English transl., Theory Probab. Math. Statist. 36 (1988), 97–110. MR 913723
- Jürgen Potthoff, Sample properties of random fields. II. Continuity, Commun. Stoch. Anal. 3 (2009), no. 3, 331–348. MR 2604006, DOI 10.31390/cosa.3.3.02
- Yiqian Shi and Bin Xu, Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary, Ann. Global Anal. Geom. 38 (2010), no. 1, 21–26. MR 2657840, DOI 10.1007/s10455-010-9198-0
- Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983), no. 1, 48–79. MR 705991, DOI 10.1016/0022-1236(83)90090-3
- Xiangjin Xu, Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier theorem, Forum Math. 21 (2009), no. 3, 455–476. MR 2526794, DOI 10.1515/FORUM.2009.021
- A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. I, Springer Series in Statistics, Springer-Verlag, New York, 1987. Basic results. MR 893393
Additional Information
- Zachary A. Gelbaum
- Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
- Email: zachgelbaum@gmail.com
- Received by editor(s): July 26, 2012
- Received by editor(s) in revised form: November 17, 2012
- Published electronically: April 1, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 4781-4814
- MSC (2010): Primary 60G60, 60G15, 58J35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06106-0
- MathSciNet review: 3217700
Dedicated: In loving memory of my grandfather, B.R. Gelbaum