Statistics of nonlinear stochastic dynamical systems under Lévy noises by a convolution quadrature approach

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar nonlinear systems excited by Lévy white noises. The proposed numerical procedure relies on the introduction of an integral transform of the Wiener–Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: (1) statistics of systems with several different drift terms can be handled in an efficient way, independently from the kind of white noise; (2) the particular form of Wiener–Hopf integral transform and its numerical evaluation, both introduced in this study, are generalizations of fractional integro-differential operators of potential type and Grünwald–Letnikov fractional derivatives, respectively.