Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Brownian sheet images and Bessel-Riesz capacity


Author: Davar Khoshnevisan
Journal: Trans. Amer. Math. Soc. 351 (1999), 2607-2622
MSC (1991): Primary 60J45; Secondary 60G15
Published electronically: February 9, 1999
MathSciNet review: 1638246
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the image of a 2-dimensional set under $d$-dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ($d/2$)-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.


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

  • 1. Robert J. Adler, The geometry of random fields, John Wiley & Sons, Ltd., Chichester, 1981. Wiley Series in Probability and Mathematical Statistics. MR 611857
  • 2. H. Ben Saud and K. Jenßen, A characterization of parabolic potential theory, Math. Ann., 272 (1985), 281-289.
  • 3. Itai Benjamini, Robin Pemantle, and Yuval Peres, Martin capacity for Markov chains, Ann. Probab. 23 (1995), no. 3, 1332–1346. MR 1349175
  • 4. N. N. C\v{e}ntsov, Wiener random fields depending on several parameters, Dokl. Akad. Nauk S.S.S.R. (NS), 106 (1956), 607-609.
  • 5. Robert C. Dalang and John B. Walsh, Local structure of level sets of the Brownian sheet, Stochastic analysis: random fields and measure-valued processes (Ramat Gan, 1993/1995) Israel Math. Conf. Proc., vol. 10, Bar-Ilan Univ., Ramat Gan, 1996, pp. 57–64. MR 1415187
  • 6. Peter Imkeller, Two-parameter martingales and their quadratic variation, Lecture Notes in Mathematics, vol. 1308, Springer-Verlag, Berlin, 1988. MR 947545
  • 7. Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
  • 8. Robert Kaufman and Jang Mei Wu, Parabolic potential theory, J. Differential Equations 43 (1982), no. 2, 204–234. MR 647063, 10.1016/0022-0396(82)90091-2
  • 9. D. Khoshnevisan Some polar sets for the Brownian sheet, Sém. de Prob., XXXI, Lecture Notes in Mathematics, vol. 1655, pp. 190-197, 1997. CMP 98:03
  • 10. D. Khoshnevisan and Z. Shi, Brownian sheet and capacity, Preprint, 1997
  • 11. Steven Orey and William E. Pruitt, Sample functions of the 𝑁-parameter Wiener process, Ann. Probability 1 (1973), no. 1, 138–163. MR 0346925
  • 12. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 13. Y. Xiao, Hitting probabilities and polar sets for fractional Brownian motion, Preprint, 1997.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 60J45, 60G15

Retrieve articles in all journals with MSC (1991): 60J45, 60G15


Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: davar@math.utah.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02408-3
Keywords: Capacity, Brownian sheet, additive Brownian motion, multi-parameter martingales.
Received by editor(s): September 23, 1997
Received by editor(s) in revised form: June 11, 1998
Published electronically: February 9, 1999
Additional Notes: Research supported by grants from the National Science Foundation and the National Security Agency
Article copyright: © Copyright 1999 American Mathematical Society