On exact time-averages of a massive Poisson particle

TL;DR

This paper derives exact long-term statistical properties of a damped oscillatory particle driven by heterogeneous Poisson noise, revealing insights into energy dynamics and stochastic resonance phenomena.

Contribution

It provides the first exact time-average analysis of a massive particle under heterogeneous Poisson noise, including equilibrium distributions and energy asymptotics.

Findings

Derived exact equilibrium distributions of position and velocity.

Obtained analytical expressions for energy injection and dissipation.

Discussed conditions for stochastic resonance emergence.

Abstract

In this work we study, under the Stratonovich definition, the problem of the damped oscillatory massive particle subject to a heterogeneous Poisson noise characterised by a rate of events, \lambda (t), and a magnitude, \Phi, following an exponential distribution. We tackle the problem by performing exact time-averages over the noise in a similar way to previous works analysing the problem of the Brownian particle. From this procedure we obtain the long-term equilibrium distributions of position and velocity as well as analytical asymptotic expressions for the injection and dissipation of energy terms. Considerations on the emergence of stochastic resonance in this type of system are also set forth.