  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 \alpha \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 $\alpha$. 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 $\alpha$ 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.
