Analytical Survival Analysis of the Ornstein–Uhlenbeck Process

• Published in: Journal of statistical physics Vol. 181; no. 6; pp. 2404 - 2414
• Main Authors: Giorgini, L. T., Moon, W., Wettlaufer, J. S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2020; Springer
• Subjects: Analysis; Asymptotics; Boundary layer; Differential equations; Fokker– Fokker–Planck equation; Mathematical and Computational Physics; Ornstein– Ornstein–Uhlenbeck Process; Physical Chemistry; Physics; Physics and Astronomy; Planck equation; Quantum Physics; Statistical Physics and Dynamical Systems; Stochastic processes; Survival probability; Theoretical; Uhlenbeck Process; Ornstein–Uhlenbeck Process; Survival probability; Fokker–Planck equation; Asymptotics
• ISSN: 0022-4715; 1572-9613; 1572-9613
• DOI: 10.1007/s10955-020-02669-y

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein–Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in boundary layers, near the lateral boundaries of the domain. The analytic solution agrees extremely well with the numerical solution and takes into account the non-negligible leakage of probability that occurs at short times when the stochastic process begins close to one of the boundaries. Given the range of applications of Ornstein–Uhlenbeck processes, the analytic solution is of broad relevance across many fields of natural and engineering science.