Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments

In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem and related concepts, the solution of the Kolmogorov-Feller equation is obtained in a very easy way by solving a set of linear differential equations. Results are compared with those of Monte Carlo simulation showing the robustness of the solution pursued in terms of CFMs. Further, a very elegant strategy, to give a relationship between the CFMs evaluated for different value of real part, is introduced as well.