Sets avoided by Brownian motion. (English) Zbl 0934.60016

Summary: 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 paper 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.


60D05 Geometric probability and stochastic geometry
60J65 Brownian motion
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