Asymptotic velocity of one dimensional diffusions with periodic drift

TL;DR

This paper investigates the long-term behavior of one-dimensional stochastic differential equations with periodic, zero-mean drift, revealing that such systems often exhibit a non-zero asymptotic velocity.

Contribution

It demonstrates that in several models, despite zero average background, the solutions tend to have a non-zero asymptotic velocity, a novel insight into periodic stochastic systems.

Findings

Non-zero asymptotic velocity in models with zero-mean periodic drift

Generically observed in several classes of one-dimensional diffusions

Challenges the expectation of zero velocity in zero-mean backgrounds

Abstract

We consider the asymptotic behaviour of the solution of one dimensional stochastic differential equations and Langevin equations in periodic backgrounds with zero average. We prove that in several such models, there is generically a non vanishing asymptotic velocity, despite of the fact that the average of the background is zero.