Computing probability density of the first passage time for state transition in stochastic dynamical systems driven by Brownian motions: A singular integral method

Nonlinear dynamical systems, such as climate systems, often switch from one metastable state to another when subject to noise. The first occurrence of such state transition, which is usually characterized by the first passage time, has gained enormous interest in many engineering and scientific fields. We develop an efficient numerical method to compute the probability density of the first passage time for state transitions in stochastic dynamical systems driven by Brownian motions. The proposed method involves solving a singular integral equation, which determines probability density of the first passage time. Some numerical examples, with application to a simplified thermohaline circulation system, are provided to illustrate and verify the proposed method.