the distribution of exit times for weakly colored noise View Full Text


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Article Info

DATE

1989-03

AUTHORS

Patrick S. Hagan, Charles R. Doering, C. David Levermore

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, whereξ(t) is Gaussian white noise. Let the exit timetex be the first time the particle escapes an interval −At} 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/A1) this solution is andF=1 otherwise. Here, α=∥ξ(1/2)∥=1.4603⋯, where ξ is the Riemann zeta function, and the constant κ is 0.22749⋯. More... »

PAGES

1321-1352

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf01044718

DOI

http://dx.doi.org/10.1007/bf01044718

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1015406253


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