Asymptotic behavior of the mean square displacement of the Brownian parametric oscillator near the singular point

TL;DR

This paper derives an analytical asymptotic formula for the long-time mean square displacement of a damped Brownian parametric oscillator near a singular point, applicable to general periodic functions, and proposes functions that reduce MSD.

Contribution

It provides the first analytical asymptotic expression for MSD of a damped Brownian parametric oscillator near a singular point, extending previous work to general periodic functions.

Findings

Derived an asymptotic formula for MSD near the singular point

Proposed periodic functions that significantly reduce MSD

Applicable to a wide class of periodic driving functions

Abstract

A parametric oscillator with damping driven by white noise is studied. The mean square displacement (MSD) in the long-time limit is derived analytically for the case that the static force vanishes, which was not treated in the past work \cite{tashiro07}. The formula is asymptotic but is applicable to a general periodic function. On the basis of this formula, some periodic functions reducing MSD remarkably are proposed.