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Flat points of two-dimensional Brownian motion
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  • Published: June 1992

Flat points of two-dimensional Brownian motion

  • Michio Shimura1 

Probability Theory and Related Fields volume 93, pages 197–229 (1992)Cite this article

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Summary

SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt 0 and δ>0 such thatZ(t 0) is an extremal point of the convex hull of {Z(t)|t 0−δ≦t≦t0} and also an extremal point of the convex hull of {Z(t)|t 0≦t≦t0+δ} and, moreover, the tangent lines to the convex hulls atZ(t 0) form a non-zero angle.

The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t 0 and δ>0 such that the intersection of the convex hulls of {Z(t)|t 0−δ≦t≦t0} and {Z(t)|t 0≦t≦t0+δ} contains onlyZ(t 0)?

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Author information

Authors and Affiliations

  1. Faculty of Science, Toho University, Miyama 2-2-1, 274, Funabashi, Chba, Japan

    Michio Shimura

Authors
  1. Michio Shimura
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Additional information

This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture

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Shimura, M. Flat points of two-dimensional Brownian motion. Probab. Th. Rel. Fields 93, 197–229 (1992). https://doi.org/10.1007/BF01195229

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  • Received: 20 March 1991

  • Revised: 16 January 1992

  • Issue Date: June 1992

  • DOI: https://doi.org/10.1007/BF01195229

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Keywords

  • Hull
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Convex Hull
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