Statistics of non-linear stochastic dynamical systems under L\'evy noises by a convolution quadrature approach

TL;DR

This paper introduces a new numerical method combining Wiener-Hopf integral transforms and convolution quadrature to efficiently compute the statistics of non-linear stochastic systems driven by Lévy noise, generalizing fractional operators.

Contribution

It presents a novel numerical approach that extends convolution quadrature to handle Wiener-Hopf transforms for Lévy noise-driven systems, enabling efficient analysis of complex non-linear stochastic dynamics.

Findings

Efficient computation of non-stationary response statistics for systems with Lévy noise.

Generalization of fractional integro-differential operators within the numerical framework.

Applicable to systems with multiple drift terms regardless of noise type.

Abstract

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral transform of 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 Gr\"unwald-Letnikov fractional derivatives, respectively.