The Annals of Probability

Sets avoided by Brownian motion

Omer Adelman, Krzysztof Burdzy, and Robin Pemantle

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A fixed two-dimensional projection of a three-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the two-dimensional projections with probability 1? Equivalently: three-dimensional Brownian motion hits any infinite cylinder with probability 1; does it hit all cylinders? This papers shows that the answer is no. Brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones.

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Ann. Probab. Volume 26, Number 2 (1998), 429-464.

First available in Project Euclid: 23 September 2004

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J65: Brownian motion [See also 58J65]

Brownian motion recurrence second moment method hitting probabilities


Adelman, Omer; Burdzy, Krzysztof; Pemantle, Robin. Sets avoided by Brownian motion. Ann. Probab. 26 (1998), no. 2, 429--464. doi:10.1214/aop/1022855639.

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