  We discuss the origin of the optical variations in the Narrow line Seyfert 1
galaxy NGC 4051 and present the results of a cross-correlation study using
X-ray and optical light curves spanning more than 12 years. The emission is
highly variable in all wavebands, and the amplitude of the optical variations
is found to be smaller than that of the X-rays, even after correcting for the
contaminating host galaxy flux falling inside the photometric aperture. The
optical power spectrum is best described by an unbroken power law model with
slope $\alpha=1.4^{+0.6}_{-0.2}$ and displays lower variability power than the
2-10 keV X-rays on all time-scales probed. We find the light curves to be
significantly correlated at an optical delay of $1.2^{+1.0}_{-0.3}$ days behind
the X-rays. This time-scale is consistent with the light travel time to the
optical emitting region of the accretion disc, suggesting that the optical
variations are driven by X-ray reprocessing. We show, however, that a model
whereby the optical variations arise from reprocessing by a flat accretion disc
cannot account for all the optical variability. There is also a second
significant peak in the cross-correlation function, at an optical delay of
$39^{+2.7}_{-8.4}$ days. The lag is consistent with the dust sublimation radius
in this source, suggesting that there is a measurable amount of optical flux
coming from the dust torus. We discuss the origin of the additional optical
flux in terms of reprocessing of X-rays and reflection of optical light by the
dust.
