Images of the Brownian sheet
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- by Davar Khoshnevisan and Yimin Xiao PDF
- Trans. Amer. Math. Soc. 359 (2007), 3125-3151 Request permission
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
- 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, Correction to: “The uniform dimension of the level sets of a Brownian sheet” [Ann. Probab. 6 (1978), no. 3, 509–515; MR 80a:60102], Ann. Probab. 8 (1980), no. 5, 1001–1002. MR 600348
- Robert J. Adler, The uniform dimension of the level sets of a Brownian sheet, Ann. Probab. 6 (1978), no. 3, 509–515. MR 490818
- R. Cairoli and John B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), 111–183. MR 420845, DOI 10.1007/BF02392100
- Jack Cuzick and Johannes P. DuPreez, Joint continuity of Gaussian local times, Ann. Probab. 10 (1982), no. 3, 810–817. MR 659550
- W. Ehm, Sample function properties of multiparameter stable processes, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 2, 195–228. MR 618272, DOI 10.1007/BF00535741
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. MR 556414
- John Hawkes, Local properties of some Gaussian processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 4, 309–315. MR 458559, DOI 10.1007/BF00533085
- Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
- Jean-Pierre Kahane, Ensembles aléatoires et dimensions, Recent progress in Fourier analysis (El Escorial, 1983) North-Holland Math. Stud., vol. 111, North-Holland, Amsterdam, 1985, pp. 65–121 (French). MR 848143, DOI 10.1016/S0304-0208(08)70281-0
- J.-P. Kahane, Ensembles parfaits et processus de Lévy, Period. Math. Hungar. 2 (1972), 49–59 (French). MR 329050, DOI 10.1007/BF02018651
- R. Kaufman, On the sum of two Brownian paths, Studia Math. 65 (1979), no. 1, 51–54. MR 554540, DOI 10.4064/sm-65-1-51-54
- Robert Kaufman, Fourier analysis and paths of Brownian motion, Bull. Soc. Math. France 103 (1975), no. 4, 427–432 (English, with French summary). MR 397905, DOI 10.24033/bsmf.1809
- Davar Khoshnevisan, Multiparameter processes, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002. An introduction to random fields. MR 1914748, DOI 10.1007/b97363
- Davar Khoshnevisan, Brownian sheet images and Bessel-Riesz capacity, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2607–2622. MR 1638246, DOI 10.1090/S0002-9947-99-02408-3
- Davar Khoshnevisan and Yimin Xiao, Lévy processes: capacity and Hausdorff dimension, Ann. Probab. 33 (2005), no. 3, 841–878. MR 2135306, DOI 10.1214/009117904000001026
- 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), no. 2, 179–185. MR 976063, DOI 10.1112/blms/21.2.179
- T. S. Mountford, An extension of a result of Kahane using Brownian local times of intersection, Stochastics 23 (1988), no. 4, 449–464. MR 943815, DOI 10.1080/17442508808833504
- Yuval Peres, Probability on trees: an introductory climb, Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 193–280. MR 1746302, DOI 10.1007/978-3-540-48115-7_{3}
- Loren D. Pitt, Local times for Gaussian vector fields, Indiana Univ. Math. J. 27 (1978), no. 2, 309–330. MR 471055, DOI 10.1512/iumj.1978.27.27024
- L. S. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 0201557
- Robert L. Wolpert, Local time and a particle picture for Euclidean field theory, J. Functional Analysis 30 (1978), no. 3, 341–357. MR 518340, DOI 10.1016/0022-1236(78)90062-9
- Yimin Xiao, Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Related Fields 109 (1997), no. 1, 129–157. MR 1469923, DOI 10.1007/s004400050128
Additional Information
- Davar Khoshnevisan
- Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, Utah 84112–0090
- MR Author ID: 302544
- 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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2299449