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research-article

Path Integral method for nonlinear systems under Levy white noise

[+] Author and Article Information
Alberto Di Matteo

Dipartimento di Ingegneria Civile, Ambientale Aerospaziale e dei Materiali (DICAM) Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy
alberto.dimatteo@unipa.it

Antonina Pirrotta

Dipartimento di Ingegneria Civile, Ambientale Aerospaziale e dei Materiali (DICAM) Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, ItalyDepartment of Mathematical Sciences, University of Liverpool, Liverpool, UK
Antonina.Pirrotta@liverpool.ac.uk

1Corresponding author.

ASME doi:10.1115/1.4036703 History: Received June 22, 2016; Revised September 21, 2016

Abstract

In this paper the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The Path Integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, Path Integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein-Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

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