Mean Exit times for Particles Driven by Weakly Colored Noise

Consider a particle whose velocity is an Ornstein-Uhlenbeck process with correlation time ε ², \[ dX/dt = σ Z/ε , dZ/dt = - Z/ε ^2 + √ 2 ξ ( t )/ε , \] where ξ ( t ) is Gaussian white noise. This represents Brownian motion with memory, i.e., a non-Markovian Brownian motion. Singular perturbation techniques reduce the mean exit time problem to a boundary layer problem, which turns out to be the half-range expansion problem for the Fokker–Planck operator. This is solved via complex variable theory. It is found that the spatial density has a Milne extrapolation length ε σ | ζ ( 1/2 ) | = ε σ ( 1.46035 ⋯ ) where ζ is the Riemann zeta function. It is also found that the average time at which $X(t)$ first escapes an interval $-A< 0 < B$ ven that it starts at $X( 0 ) = 0$ andZ (0)= z₀ is \[ E\{tₑₓ\} = 1/2σ^2 (A+ε σ |ζ (1/2)| + εσ z_0)(B+ ε σ |ξ (1/2)|-εσ z_0)+ε^2κ, \] within a transcendentally smal O( e- σ /ε σ + e- B/ε σ ) error. Here the constant κ is .22749 ⋯. Analogous first passage time formulas are obtained for particles whose velocity is an n-level Markov process.