ISSN:
1572-9613
Keywords:
Exit times
;
first passage times
;
colored noise
;
Ornstein-Uhlenbeck process
;
half-range expansion
;
singular perturbation methods
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract We analyze the exit time (first passage time) problem for the Ornstein-Uhlenbeck model of Brownian motion. Specifically, consider the positionX(t) of a particle whose velocity is an Ornstein-Uhlenbeck process with amplitudeσ/ρ and correlation time ε2, $$dX/dt = \sigma Z/\varepsilon , dZ/dt = - Z/\varepsilon ^2 + 2^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \xi (t)/\varepsilon $$ whereξ(t) is Gaussian white noise. Let the exit timet ex be the first time the particle escapes an interval −A〈X(t)〈B, given that it starts atX(0)=0 withZ(0)=z 0. Here we determine the exit time probability distributionF(t)≡Prob {t ex〉t} by directly solving the Fokker-Planck equation. In brief, after taking a Laplace transform, we use singular perturbation methods to reduce the Fokker-Planck equation to a boundary layer problem. This boundary layer problem turns out to be a half-range expansion problem, which we solve via complex variable techniques. This yields the Laplace transform ofF(t) to within a transcendentally smallO(e −A/εσ +e −B/εσ error. We then obtainF(t) by inverting the transform order by order in ε. In particular, by lettingB→∞ we obtain the solution to Wang and Uhlenbeck's unsolved problem b; throughO(ε2σ2/A 1) this solution is $$F(t) = Erf\left\{ {\frac{{A + \varepsilon \sigma \alpha + \varepsilon \sigma z_0 }}{{2\sigma (t - \varepsilon ^2 \kappa )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right\} + ... for \frac{t}{{\varepsilon ^2 }} 〉 〉 1$$ andF=1 otherwise. Here, α=∥ξ(1/2)∥=1.4603⋯, where ξ is the Riemann zeta function, and the constant κ is 0.22749⋯.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01044718
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