TY - JOUR
T1 - Asymptotic solution of the Wang-Uhlenbeck recurrence time problem
AU - Singer, A.
AU - Schuss, Z.
PY - 2005/9/9
Y1 - 2005/9/9
N2 - 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.
AB - 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.
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U2 - 10.1103/PhysRevLett.95.110601
DO - 10.1103/PhysRevLett.95.110601
M3 - Article
C2 - 16196994
AN - SCOPUS:27144462611
SN - 0031-9007
VL - 95
JO - Physical review letters
JF - Physical review letters
IS - 11
M1 - 110601
ER -