  We study the large deviations of the time-integrated current for a driven
diffusion on the circle, often used as a model of nonequilibrium systems. We
obtain the large deviation functions describing the current fluctuations using
a Fourier-Bloch decomposition of the so-called tilted generator and also
construct from this decomposition the effective (biased, auxiliary or driven)
Markov process describing the diffusion as current fluctuations are observed in
time. This effective process provides a clear physical explanation of the
various fluctuation regimes observed. It is used here to obtain an upper bound
on the current large deviation function, which we compare to a recently-derived
entropic bound, and to study the low-noise limit of large deviations.
