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Fractional Brownian fields over manifolds


Author: Zachary A. Gelbaum
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
Published electronically: April 1, 2014
MathSciNet review: 3217700
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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.


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  • [1] Robert J. Adler, The geometry of random fields, John Wiley & Sons Ltd., Chichester, 1981. Wiley Series in Probability and Mathematical Statistics. MR 611857 (82h:60103)
  • [2] Robert J. Adler and Jonathan E. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516 (2008m:60090)
  • [3] A. Benassi, Locally self-similar Gaussian processes, Wavelets and statistics (Villard de Lans, 1994) Lecture Notes in Statist., vol. 103, Springer, New York, 1995, pp. 43-54. MR 1364671 (96m:60097), https://doi.org/10.1007/978-1-4612-2544-7_4
  • [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 (98k:60056), https://doi.org/10.4171/RMI/217
  • [5] Albert Benassi and Daniel Roux, Elliptic self-similar stochastic processes, Rev. Mat. Iberoamericana 19 (2003), no. 3, 767-796. MR 2053563 (2004m:60085), https://doi.org/10.4171/RMI/369
  • [6] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin, 1971 (French). MR 0282313 (43 #8025)
  • [7] 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 (86g:58140)
  • [8] Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187 (2002h:58040)
  • [9] 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 Geom. 17 (1982), no. 1, 15-53. MR 658471 (84b:58109)
  • [10] 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 (82i:58065), https://doi.org/10.1002/cpa.3160340404
  • [11] 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 (90k:58214)
  • [12] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR 1103113 (92a:35035)
  • [13] 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 (94f:58121), https://doi.org/10.1112/blms/25.3.289
  • [14] 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 (85e:58140), https://doi.org/10.1512/iumj.1983.32.32046
  • [15] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31 #6062)
  • [16] 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 (35 #6172)
  • [17] A. 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), Vol. II: Probability theory (Berkeley, Calif.), Univ. California Press, 1972, pp. 369-374. MR 0410880 (53 #14623)
  • [18] Alexander Grigor'yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), no. 1, 33-52. MR 1443330 (98g:58167)
  • [19] Alexander Grigor'yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI, 2009. MR 2569498 (2011e:58041)
  • [20] 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 (electronic). MR 1855636 (2002j:53031), https://doi.org/10.1090/S0002-9939-01-06056-7
  • [21] Jacques Istas, Spherical and hyperbolic fractional Brownian motion, Electron. Comm. Probab. 10 (2005), 254-262 (electronic). MR 2198600 (2008b:60075), https://doi.org/10.1214/ECP.v10-1166
  • [22] Jacques Istas, On fractional stable fields indexed by metric spaces, Electron. Comm. Probab. 11 (2006), 242-251 (electronic). MR 2266715 (2008b:60076), https://doi.org/10.1214/ECP.v11-1223
  • [23] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726 (99f:60082)
  • [24] 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 (98j:58109)
  • [25] Paul Lévy, Processus stochastiques et mouvement brownien, Suivi d'une note de M. Loève. Deuxième édition revue et augmentée, Gauthier-Villars & Cie, Paris, 1965 (French). MR 0190953 (32 #8363)
  • [26] 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 (87k:58259), https://doi.org/10.2307/1971385
  • [27] 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 0266100 (42 #1009)
  • [28] 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 (89b:60194)
  • [29] Jürgen Potthoff, Sample properties of random fields. II. Continuity, Commun. Stoch. Anal. 3 (2009), no. 3, 331-348. MR 2604006 (2011a:60186)
  • [30] 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 (2011f:58051), https://doi.org/10.1007/s10455-010-9198-0
  • [31] Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), no. 1, 48-79. MR 705991 (84m:58138), https://doi.org/10.1016/0022-1236(83)90090-3
  • [32] 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 (2010d:58026), https://doi.org/10.1515/FORUM.2009.021
  • [33] 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 (89a:60105)

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Additional Information

Zachary A. Gelbaum
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: zachgelbaum@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06106-0
Received by editor(s): July 26, 2012
Received by editor(s) in revised form: November 17, 2012
Published electronically: April 1, 2014
Dedicated: In loving memory of my grandfather, B.R. Gelbaum
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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