Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A. Singer, Z. Schuss

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish (US)
Article number110601
JournalPhysical review letters
Issue number11
StatePublished - Sep 9 2005
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Physics and Astronomy


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