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Remark on exit times from cones in \(\mathbb{R}^n \)of Brownian motion
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  • Published: September 1988

Remark on exit times from cones in \(\mathbb{R}^n \)of Brownian motion

Prob. Th. Rel. Fields 74, 1–29 (1987)

  • R. Dante De Blassie1 

Probability Theory and Related Fields volume 79, pages 95–97 (1988)Cite this article

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Summary

Our purpose is to show how the asymptotics in Corollary 1.3 of [2] can be obtained under much weaker hypotheses. It turns out the problem essentially reduces to showing that if R(s) is a Bessel process, u>0 and α>0, then

$$P\left( {\mathop f\limits_0^1 R(s)^{ - 2} ds \leqq u} \right) = O(t^{ - \alpha } )$$

as t→∞. We provide a simple proof of this fact.

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References

  1. Burkholder, D.L.: Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math. 26, 182–205 (1977)

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  2. DeBlassie, R.D.: Exit times from cones in \(\mathbb{R}^n \) of Brownian motion. Probab. Th. Rel. Fields 74, 1–29 (1987)

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  3. Itô, K., McKean, H.P.: Diffusion processes and their sample paths, 2nd edn. Berlin Heidelberg New York: Springer 1974

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  4. Port, S.C., Stone, C.J.: Brownian motion and classical potential theory. New York: Academic Press 1978

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Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    R. Dante De Blassie

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  1. R. Dante De Blassie
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Supported in part by NSF

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Cite this article

De Blassie, R.D. Remark on exit times from cones in \(\mathbb{R}^n \)of Brownian motion. Probab. Th. Rel. Fields 79, 95–97 (1988). https://doi.org/10.1007/BF00319106

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  • Received: 15 May 1987

  • Revised: 10 March 1988

  • Issue Date: September 1988

  • DOI: https://doi.org/10.1007/BF00319106

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Keywords

  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Statistical Theory
  • Simple Proof
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