Singular probability distribution of shot-noise driven systems

TL;DR

This paper analyzes the stationary probability distribution of systems driven by shot noise, revealing power-law singularities and peak structures in both overdamped and underdamped regimes, supported by analytical and numerical results.

Contribution

It provides a detailed analytical characterization of the probability distribution's singularities and peak positions in shot-noise driven systems, including geometric progression patterns.

Findings

Distribution exhibits power-law singularities in the central part.

Peaks in the distribution are characterized by specific exponents.

Peak positions follow a geometric progression in the underdamped regime.

Abstract

We study the stationary probability distribution of a system driven by shot noise. We find that, both in the overdamped and underdamped regime, the coordinate distribution displays power-law singularities in its central part. For sufficiently low rate of noise pulses they correspond to distribution peaks. We find the positions of the peaks and the corresponding exponents. In the underdamped regime the peak positions are given by a geometric progression. The energy distribution in this case also displays multiple peaks with positions given by a geometric progression. Such structure is a signature of the shot-noise induced fluctuations. The analytical results are in excellent agreement with numerical simulations.