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Transactions of the American Mathematical Society

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Images of the Brownian sheet


Authors: Davar Khoshnevisan and Yimin Xiao
Journal: Trans. Amer. Math. Soc. 359 (2007), 3125-3151
MSC (2000): Primary 60G15, 60G17, 28A80
DOI: https://doi.org/10.1090/S0002-9947-07-04073-1
Published electronically: February 14, 2007
MathSciNet review: 2299449
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Abstract: An $ N$-parameter Brownian sheet in $ \mathbf{R}^d$ maps a non-random compact set $ F$ in $ \mathbf{R}^N_+$ to the random compact set $ B(F)$ in $ \mathbf{R}^d$. We prove two results on the image-set $ B(F)$:

(1) It has positive $ d$-dimensional Lebesgue measure if and only if $ F$ has positive $ \frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999).

(2) If $ \dim_{_\mathcal{H}}F > \frac d 2$, then with probability one, we can find a finite number of points $ \zeta_1,\ldots,\zeta_m\in\mathbf{R}^d$ such that for any rotation matrix $ \theta$ that leaves $ F$ in $ \mathbf{R}^N_+$, one of the $ \zeta_i$'s is interior to $ B(\theta F)$. In particular, $ B(F)$ has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989).

This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).'' Both ideas may be of independent interest.

We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).


References [Enhancements On Off] (What's this?)

  • [1] Robert J. Adler, The Geometry of Random Fields, Wiley, 1981. MR 0611857 (82h:60103)
  • [2] Robert J. Adler, Correction to: ``The uniform dimension of the level sets of a Brownian sheet'' [Ann. Probab. 6 (1978), no. 3, 509-515], Ann. Probab. 8(5) (1980), 1001-1002. MR 0600348 (82h:60148)
  • [3] Robert J. Adler, The uniform dimension of the level sets of a Brownian sheet, Ann. Probab. 6(3) (1978), 509-515. MR 0490818 (80a:60102)
  • [4] R. Cairoli and John B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1977), 111-183. MR 0420845 (54:8857)
  • [5] Jack Cuzick and Johannes P. DuPreez, Joint continuity of Gaussian local times, Ann. Probab. 10(3) (1982), 810-817. MR 0659550 (83f:60061)
  • [6] W. Ehm, Sample function properties of multiparameter stable processes, Z. Wahrsch. Verw. Gebiete 56(2) (1981), 195-228. MR 0618272 (82g:60096)
  • [7] Kenneth Falconer, Fractal Geometry, John Wiley & Sons Ltd. 1990. MR 1102677 (92j:28008)
  • [8] Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8(1) (1980), 1-67. MR 0556414 (81b:60076)
  • [9] John Hawkes, Local properties of some Gaussian processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40(4) (1977), 309-315. MR 0458559 (56:16759)
  • [10] Jean-Pierre Kahane, Some Random Series of Functions, Second Edition, Cambridge University Press, 1985a. MR 0833073 (87m:60119)
  • [11] Jean-Pierre Kahane, Ensembles aléatoires et dimensions, In: Recent progress in Fourier analysis (El Escorial, 1983), 65-121, North-Holland, Amsterdam (1985b). MR 0848143 (87k:60110)
  • [12] Jean-Pierre Kahane, Ensembles parfaits et processus de Lévy, Period. Math. Hungar. 2 (1972), 49-59. MR 0329050 (48:7392)
  • [13] Robert Kaufman, On the sum of two Brownian paths, Studia. Math. LXV (1979), 51-54. MR 0554540 (80k:60100)
  • [14] Robert Kaufman, Fourier analysis and paths of Brownian motion, Bull. Soc. Math. France 103 (1975), 427-432. MR 0397905 (53:1760)
  • [15] Davar Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields, Springer, 2002. MR 1914748 (2004a:60003)
  • [16] Davar Khoshnevisan, Brownian sheet images and Bessel-Riesz capacity, Trans. Amer. Math. Soc. 351(7) (1999), 2607-2622. MR 1638246 (2000e:60123)
  • [17] Davar Khoshnevisan and Yimin Xiao, Lévy Processes: Capacity and Hausdorff dimension, Ann. Probab. 33(3) (2005), 841-878. MR 2135306 (2006d:60078)
  • [18] T. S. Mountford, A relation between Hausdorff dimension and a condition on time sets for the image by the Brownian sheet to possess interior-points, Bull. London Math. Soc. 21 (1989), 179-185. MR 0976063 (89m:60094)
  • [19] T. S. Mountford, An extension of a result of Kahane using Brownian local times of intersection, Stochastics 23(4) (1988), 449-464. MR 0943815 (89m:60197)
  • [20] Yuval Peres, Probability on Trees: An Introductory Climb, Lectures on probability theory and statistics (Saint-Flour, 1997), 1999, pp. 193-280. MR 1746302 (2001c:60139)
  • [21] Lauren D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), 309-330. MR 0471055 (57:10796)
  • [22] L. S. Pontryagin, Topological Groups, Translated from the second Russian edition by Arlen Brown, Gordon and Breach Science Publishers, Inc. 1966. MR 0201557 (34:1439)
  • [23] Robert L. Wolpert, Local time and a particle picture for Euclidean field theory, J. Funct. Anal. 30(3) (1978), 341-357. MR 0518340 (80a:81082)
  • [24] Yimin Xiao, Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Rel. Fields 109 (1997), 129-157. MR 1469923 (98m:60060)

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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, Utah 84112–0090
Email: davar@math.utah.edu

Yimin Xiao
Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824
Email: xiao@stt.msu.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04073-1
Keywords: Brownian sheet, image, Bessel--Riesz capacity, Hausdorff dimension, interior-point
Received by editor(s): September 12, 2004
Received by editor(s) in revised form: April 21, 2005
Published electronically: February 14, 2007
Additional Notes: This research was supported by a generous grant from the National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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