# Underdamped stochastic harmonic oscillator

## TL;DR

This paper studies the stationary states of a damped stochastic harmonic oscillator driven by Lévy noise, revealing non-equipartition energy distributions, a damping-controlled energy partition, and universal power-law energy ratios.

## Contribution

It introduces a stochastic analogue of the equipartition theorem for systems driven by Lévy noise and analyzes energy distribution asymptotics in the vanishing damping limit.

## Key findings

- Energy distributions follow power-law asymptotics.
- Partition of energy depends on damping coefficient.
- Energy ratio exhibits universal power-law behavior.

## Abstract

We investigate stationary states of the linear damped stochastic oscillator driven by L\'evy noises. In the long time limit kinetic and potential energies of the oscillator do not fulfill the equipartition theorem and their distributions follow the power-law asymptotics. At the same time, partition of the mechanical energy is controlled by the damping coefficient. We show that in the limit of vanishing damping a stochastic analogue of the equipartition theorem can be proposed, namely the statistical properties of potential and kinetic energies attain distributions characterized by the same width. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics.