The distribution of exit times for weakly colored noise (Q751069)

We analyze the exit time (first passage time) problem for the Ornstein- Uhlenbeck model of Brownian motion. Specifically, consider the position X(t) of a particle whose velocity is an Ornstein-Uhlenbeck process with amplitude \(\sigma\) /\(\epsilon\) and correlation time \(\epsilon^ 2\), \[ dX/dt=\sigma Z/\epsilon,\quad dZ/dt=-Z/\epsilon^ 2+2^{1/2}\xi (t)/\epsilon \] where \(\xi\) (t) is Gaussian white noise. Let the exit time \(t_{ex}\) be the first time the particle escapes an interval \(- A<X(t)<B\), given that it starts at \(X(0)=0\) with \(Z(0)=z_ 0\). Here we determine the exit time probability distribution \(F(t)\equiv \Pr ob\{t_{ex}>t\}\) 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 of F(t) to within a transcendentally small \(O(e^{- A/\epsilon \sigma}+e^{-B/\epsilon \sigma})\) error. We then obtain F(t) by inverting the transform order by order in \(\epsilon\).