  This paper establishes the global asymptotic equivalence, in the sense of the
Le Cam $\Delta$-distance, between scalar diffusion models with unknown drift
function and small variance on the one side, and nonparametric autoregressive
models on the other side. The time horizon $T$ is kept fixed and both the cases
of discrete and continuous observation of the path are treated. We allow non
constant diffusion coefficient, bounded but possibly tending to zero. The
asymptotic equivalences are established by constructing explicit equivalence
mappings.
