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arXiv · 1411.0041

Patterns in random walks and Brownian motion

Jim Pitman·Wenpin Tang
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Abstract

We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.

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Jim Pitman, Wenpin Tang. 2014-10-31. Patterns in random walks and Brownian motion. https://doi.org/10.1007/978-3-319-18585-9_4

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