Asymptotic velocity of one dimensional diffusions with periodic drift

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.

💡 Research Summary

The paper investigates the long‑time behavior of one‑dimensional stochastic differential equations (SDEs) and Langevin equations when the driving drift field is periodic with zero spatial average. Contrary to the naive expectation that a zero‑mean periodic background should produce no net transport, the authors prove that, under generic conditions, the asymptotic velocity of the particle is non‑zero.

The analysis begins by formulating two prototypical models. The first is a pure diffusion SDE
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