Two-states Brownian particle in a Harmonic Potential

TL;DR

This paper analyzes the dynamics of a Brownian particle with randomly switching diffusion coefficients in a harmonic potential, revealing non-Gaussian distributions and cusp formation under certain conditions through analytical and numerical methods.

Contribution

It provides analytical expressions for the probability distribution, mean square displacement, and kurtosis of a two-state diffusion Brownian particle in a harmonic potential, including the case with zero diffusion.

Findings

Distribution exhibits non-Gaussian behavior in the long-term limit.

Probability distribution develops a cusp when one diffusion coefficient is zero.

Results are supported by numerical simulations.

Abstract

We study the behaviour of a Brownian particle in the overdamped regime in the presence of a harmonic potential, assuming its diffusion coefficient to randomly jump between two distinct values. In particular, we characterize the probability distribution of the particle position and provide detailed expressions for the mean square displacement and the kurtosis. We highlight non-Gaussian behaviour even within the long-term limit carried over with an excess of probability both in the central part and in the distribution's tails. Moreover, when one of the two diffusion coefficients assumes the value zero, we provide evidence that the probability distribution develops a cusp. Most of our results are analytical, and corroborated by numerical simulations.