Effective diffusion of Brownian motion in spatially quasi-periodic noise

The effective diffusion of Brownian particles in periodic potential has been a central topic in nonequilibrium statistical physcis. A classical result is the Lifson formula which provides the effective diffusion constant in periodic potentials. Extending beyong periodicity, our recent work [arXiv:2504.16527] has demonstrated that a modified Lifson expression remains valid for Brownian motion in quasi-periodic potentials. In this work, we extend our previous results by incorporating spatial quasi-periodic noise and examining different stochastic interpretations, $\alpha\in[0,1]$. The proposed framework is simple, computationally efficient, and unifies the treatment of diffusion in both periodic and quasi-periodic systems.