  It is well-known that the SRB measure of a $C^{1+\alpha}$ Anosov
diffeomorphism has exponential decay of correlations with respect to
H{\"o}lder-continuous observables. We propose a new approach to this
phenomenon, based on optimal transport. More precisely, we define a space of
measures having absolutely continuous disintegrations with respect to some
foliation close to the unstable foliation of the map, endowed with a variant of
the Wasserstein metric where mass is only allowed to be transported along the
diffeomorphism's stable foliation. We show that this metric is indeed finite on
that space, and use that the construction makes the diffeomorphism act as a
contraction to deduce two corollaries. First, the SRB measure has exponential
decay of correlation with respect to pairs of observable that are only asked to
be H{\"o}lder-continuous \emph{in the stable, respectively unstable direction},
but can be discontinuous overall. Then, we prove quantitative statistical
stability: the map sending a $C^{1+\alpha}$ Anosov diffeomorphism to its SRB
measure is locally H{\"o}lder-continuous (using the $C^1$ metric for
diffeomorphisms and the usual Wasserstein metric for measures).
