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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
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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.

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  • 1. R. J. Adler, The Geometry of Random Fields, Wiley, London, 1981. MR 82h:60103
  • 2. H. Ben Saud and K. Jenßen, A characterization of parabolic potential theory, Math. Ann., 272 (1985), 281-289.
  • 3. I. Benjamini, R. Pemantle and Y. Peres, Martin capacity for Markov chains, Ann. Prob., 23 (1995), 1332-1346. MR 96g:60098
  • 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. R. C. Dalang and J. B. Walsh, Geography of the level sets of the Brownian sheet, Prob. Th. Rel. Fields, 96 (1993), 153-176. MR 98f:60093
  • 6. P. Imkeller, Two-parameter Martingales and Their Quadratic Variation, Lecture Notes in Mathematics, vol. 1308, Springer, New York, 1988. MR 89e:60098
  • 7. J.-P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge, 1985. MR 87m:60119
  • 8. R. Kaufman and J. M. Wu, Parabolic Potential Theory, J. Diff. Eq., 43, (1982), 204-234. MR 83d:31006
  • 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. S. Orey and W. E. Pruitt, Sample functions of the $N$-parameter Wiener process, Ann. Prob., 1 (1973), 138-163. MR 49:11646
  • 12. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Fifth Edition, 1986. MR 44:7280
  • 13. Y. Xiao, Hitting probabilities and polar sets for fractional Brownian motion, Preprint, 1997.

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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

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

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