On the time for Brownian motion to visit every point on a circle

Consider a Wiener process W on a circle of circumference L. We prove the rather surprising result that the Laplace transform of the distribution of the first time, θL, when the Wiener process has visited every point of the circle can be solved in closed form using a continuous recurrence approach.

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Bibliographic Details
Published in:Journal of statistical planning and inference 2016-04, Vol.171, p.130-134
Main Authors: Ernst, Philip, Shepp, Larry
Format: Article
Language:English
Subjects:
Continuous recurrence
First hitting time
Range of Wiener process
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Summary:Consider a Wiener process W on a circle of circumference L. We prove the rather surprising result that the Laplace transform of the distribution of the first time, θL, when the Wiener process has visited every point of the circle can be solved in closed form using a continuous recurrence approach.
ISSN:0378-3758
1873-1171
DOI:10.1016/j.jspi.2015.10.010