The effective diffusion constant of stochastic processes with spatially periodic noise

We discuss the effective diffusion constant $D_{\it eff}$ for stochastic processes with spatially-dependent noise. Starting from a stochastic process given by a Langevin equation, different drift-diffusion equations can be derived depending on the choice of the discretization rule $ 0 \leq α\leq 1$. We initially study the case of periodic heterogeneous diffusion without drift and we determine a general result for the effective diffusion coefficient $D_{\it eff}$, which is valid for any value of $α$. We study the case of periodic sinusoidal diffusion in detail and we find a relationship with Legendre functions. Then, we derive $D_{\it eff}$ for general $α$ in the case of diffusion with periodic spatial noise and in the presence of a drift term, generalizing the Lifson-Jackson theorem. Our results are illustrated by analytical and numerical calculations on generic periodic choices for drift and diffusion terms.

💡 Research Summary

The paper investigates the long‑time, large‑scale transport properties of one‑dimensional stochastic processes whose diffusion coefficient is a spatially periodic function g(x). Starting from the Langevin equation
dx/dt = g(x) ξ(t)
with Gaussian white noise ξ(t), the authors emphasize that the stochastic integral must be interpreted via a discretization parameter α (0 ≤ α ≤ 1). This parameter interpolates between the Itô (α = 0), Stratonovich (α = ½) and Hänggi‑Klimontovich (α = 1) conventions. For a general α the associated Fokker‑Planck equation reads
∂ₜW = ∂ₓ