Mean Exit Times for Particles Driven by Weakly Colored Noise

• Published in: SIAM journal on applied mathematics 1989-10, Vol.49 (5), p.1480-1513
• Main Authors: Hagan, Patrick S., Doering, Charles R., Levermore, C. David
• Format: Article
• Language: English
• Subjects: Aerodynamic noise; Boundary conditions; Boundary layers; Brownian motion; Differential equations; Eigenfunctions; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; Markov processes; Mathematical extrapolation; Noise; Ornstein Uhlenbeck process; Particle density; Physics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Velocity; White noise
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0149090

Consider a particle whose velocity is an Ornstein-Uhlenbeck process with correlation time ε2, $\frac{dX}{dt} = \frac{\sigma Z}{\varepsilon},\quad \frac{dZ}{dt} = - \frac{Z}{\varepsilon^2} + \frac{\sqrt 2 \xi(t)}{\varepsilon},$ 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 $- A < 0 < B$, given that it starts at X(0) = 0 and Z(0) = z0, is E{tex} = 1/2 σ2(A + ε σ |ζ (1/2)| + ε σ z0) (B + ε σ |ζ (1/2)| - ε σ z0) + ε2 κ, to within a transcendentally small O(e-A/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.