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.
Keywords
AMS 2010 Mathematics Subject Classification: 60C05, 60G17, 60J65.
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Acknowledgements
We would like to express our gratitude to Patrick Fitzsimmons for posing the question whether one can find the distribution of Vervaat bridges by a random spacetime shift of Brownian motion. We thank Steven Evans for helpful discussion on potential theory, and Davar Koshnevisan for remarks on additive Lévy processes. We also thank an anonymous referee for his careful reading and helpful suggestions.
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Pitman, J., Tang, W. (2015). Patterns in Random Walks and Brownian Motion. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_4
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