Patterns in random walks and Brownian motion

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.

31 pages, 4 figures. This paper is published at http://link.springer.com/chapter/10.1007/978-3-319-18585-9_4

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• The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift
• The argmin process of random walks, Brownian motion and Lévy processes
• The spans in Brownian motion