Closure schemes in stochastic nonlinear dynamics: A validation case study

It is known that randomness in dynamical systems often leads to infinite hierarchies of coupled equations for relevant probabilistic quantities. Several closure methods for truncation of the hierarchies have been proposed in the literature. In the present paper the performance of closure schemes for moment hierarchies is compared by using the well-known nonlinear equation of overdamped oscillator with additive Gaussian white noise. In the case of bistable dynamics it is shown that the closure schemes can give incorrect results despite of their good convergence properties. To overcome this deficiency new closure procedures are proposed. They are based on using the Hermite polynomials and their generalization.