Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem." We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t-3/2 decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.