The Annals of Probability

Stable processes have thorns

Krzysztof Burdzy and Tadeusz Kulczycki

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Abstract

Let $X(t)$ be the symmetric $\alpha$-stable process in $\R$, $\alpha \in (0,2)$, $d \ge 2$. For $f\dvtx (0,1) \to (0,\infty)$ let $D(f)$ be the thorn $\{x \in \R\dvtx x_{1} \in (0,1),\allowbreak |(x_{2},\ldots,x_{d})| < f(x_{1})\}$. We give an integral criterion in terms of $f$ for the existence of a random time $s $ such that $X(t)$ remains in $X(s) + \overline{D}(f)$ for all $t \in [s,s+1)$.

Article information

Source
Ann. Probab. Volume 31, Number 1 (2003), 170-194.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1046294308

Digital Object Identifier
doi:10.1214/aop/1046294308

Mathematical Reviews number (MathSciNet)
MR1959790

Zentralblatt MATH identifier
1019.60035

Subjects
Primary: 60G17: Sample path properties 60G52: Stable processes

Keywords
Symmetric stable process local properties of trajectories thorn points thorns

Citation

Burdzy, Krzysztof; Kulczycki, Tadeusz. Stable processes have thorns. Ann. Probab. 31 (2003), no. 1, 170--194. doi:10.1214/aop/1046294308. http://projecteuclid.org/euclid.aop/1046294308.


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References

  • [1] ADELMAN, O., BURDZY, K. and PEMANTLE, R. (1998). Sets avoided by Brownian motion. Ann. Probab. 26 429-464.
  • [2] BASS, R. and BURDZY, K. (1999). Cutting Brownian Paths. Amer. Math. Soc., Providence, RI.
  • [3] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [4] BLUMENTHAL, R. M. and GETOOR, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.
  • [5] BLUMENTHAL, R. M., GETOOR, R. K. and RAY, D. B. (1961). On the distribution of first hits for the sy mmetric stable processes. Trans. Amer. Math. Soc. 99 540-554.
  • [6] BOGDAN, K. (1999). Representation of -harmonic functions in Lipschitz domains. Hiroshima Math. J. 29 227-243.
  • [7] BOGDAN, K., BURDZY, K. and CHEN, Z.-Q. Censored stable processes. Preprint.
  • [8] BOGDAN, K. and By CZKOWSKI, T. (1999). Potential theory for the -stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133 53-92.
  • [9] BOGDAN, K. and By CZKOWSKI, T. (1999). Probabilistic proof of boundary Harnack principle for -harmonic functions. Potential Anal. 11 135-156.
  • [10] BURDZY, K. (1985). Brownian paths and cones. Ann. Probab. 13 1006-1010.
  • [11] CHEN, Z.-Q. (1999). Multidimensional sy mmetric stable processes. Korean J. Comput. Appl. Math. 6 227-266.
  • [12] DAVIS, B. (1983). On Brownian slow points. Z. Wahrsch. Verw. Gebiete 64 359-367.
  • [13] DVORETZKY, A. (1963). On the oscillation of the Brownian motion process. Israel J. Math. 1 212-214.
  • [14] EVANS, S. N. (1985). On the Hausdorff dimension of Brownian cone points. Math. Proc. Cambridge Philos. Soc. 98 343-353.
  • [15] GREENWOOD, P. and PERKINS, E. (1983). A conditioned limit theorem for random walk and Brownian local time on square root boundaries. Ann. Probab. 11 227-261.
  • [16] IKEDA, N. and WATANABE, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Ky oto Univ. 2 79-95.
  • [17] JAFFARD, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207-227.
  • [18] LE GALL, J.-F. (1987). Mouvement brownien, cônes et processus stables. Probab. Theory Related Fields 76 587-627.
  • [19] SHIMURA, M. (1985). Excursions in a cone for two-dimensional Brownian motion. J. Math. Ky oto Univ. 25 433-443.
  • SEATTLE, WASHINGTON 98195-4350 E-MAIL: burdzy@math.washington.edu INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES UL. KOPERNIKA 18 51-617 WROCLAW POLAND AND INSTITUTE OF MATHEMATICS WROCLAW UNIVERSITY OF TECHNOLOGY UL. Wy BRZE ZE Wy SPIA´NSKIEGO 27 50-370 WROCLAW POLAND E-MAIL: tkulczy c@kac.im.pwr.wroc.pl