Path integral solution for non-linear system enforced by Poisson White Noise

Abstract

In this paper the response in terms of probability density function of non-linear systems under Poisson White Noise is considered. The problem is handled via Path Integral (PI) solution that may be considered as a step-by-step solution technique in terms of probability density function. First the extension of the PI to the case of Poisson White Noise is derived, then it is shown that at the limit when the time step becomes an infinitesimal quantity the Kolmogorov-Feller (K-F) equation is fully restored enforcing the validity of the approximations made in obtaining the conditional probability appearing in the Chapman Kolmogorov equation (starting point of the PI). Spectral counterpart of the PI, ruling the evolution of the characteristic function is also derived. It is also shown that using appropriately the PI for Poisson White Noise also the case of Normal White Noise be easily derived.