Flat points of two-dimensional Brownian motion |
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Authors: | Michio Shimura |
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Affiliation: | (1) Faculty of Science, Toho University, Miyama 2-2-1, 274 Funabashi, Chba, Japan |
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Abstract: | Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt0 and >0 such thatZ(t0) is an extremal point of the convex hull of {Z(t)|t0–tt0} and also an extremal point of the convex hull of {Z(t)|t0tt0+} and, moreover, the tangent lines to the convex hulls atZ(t0) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t0 and >0 such that the intersection of the convex hulls of {Z(t)|t0–tt0} and {Z(t)|t0tt0+} contains onlyZ(t0)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture |
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