On the Time for Brownian Motion to Visit Every Point on a Circle

Philip Ernst, Larry Shepp

Published 2013-12-17, updated 2016-05-11Version 2

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, $\theta_L$, when the Wiener process has visited every point of the circle can be solved in closed form using a continuous recurrence approach.