First Passage Time Problems With Applications to Synchronization

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First passage time problems arise in many fields, but few of them are explicitly solved. Motivated by an application to synchronization of clocks, this paper obtains the probability distribution for the first passage time for a Brownian motion with quadratic drift to exit from two constant barriers that are not necessarily symmetric. The expression for the distribution is explicit, consisting of elementary functions and functions that are characterized by Laplace transforms. Applications of first passage times are demonstrated in the context of synchronization.