Exact stationary solutions of the Kolmogorov-Feller equation in a bounded domain

TL;DR

This paper derives and analyzes the exact stationary solutions of the Kolmogorov-Feller equation for bounded jump processes driven by Poisson noise, revealing complex behaviors influenced by process parameters.

Contribution

It provides the first detailed analytical solutions for bounded jump processes with Poisson noise, highlighting how process features depend on the ratio of saturation width to pulse size.

Findings

Stationary PDFs can exhibit complex multi-branch structures.

The shape and extremal probabilities of the PDF are fully determined by process parameters.

Numerical simulations confirm the analytical solutions.

Abstract

We present the first detailed analysis of the statistical properties of jump processes bounded by a saturation function and driven by Poisson white noise, being a random sequence of delta pulses. The Kolmogorov-Feller equation for the probability density function (PDF) of such processes is derived and its stationary solutions are found analytically in the case of the symmetric uniform distribution of pulse sizes. Surprisingly, these solutions can exhibit very complex behavior arising from both the boundedness of pulses and processes. We show that all features of the stationary PDF (number of branches, their form, extreme values probability, etc.) are completely determined by the ratio of the saturation function width to the half-width of the pulse-size distribution. We verify all theoretical results by direct numerical simulations.