  We consider the problem of comparing several samples of stochastic processes
with respect to their second-order structure, and describing the main modes of
variation in this second order structure, if present. These tasks can be seen
as an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of
covariance operators, respectively. They arise naturally in functional data
analysis, where several populations are to be contrasted relative to the nature
of their dispersion around their means, rather than relative to their means
themselves. We contribute a novel approach based on optimal (multi)transport,
where each covariance can be identified with a a centred Gaussian process of
corresponding covariance. By means of constructing the optimal simultaneous
coupling of these Gaussian processes, we contrast the (linear) maps that
achieve it with the identity with respect to a norm-induced distance. The
resulting test statistic, calibrated by permutation, is seen to distinctly
outperform the state-of-the-art, and to furnish considerable power even under
local alternatives. This effect is seen to be genuinely functional, and is
related to the potential for perfect discrimination in infinite dimensions. In
the event of a rejection of the null hypothesis stipulating equality, a
geometric interpretation of the transport maps allows us to construct a
(tangent space) PCA revealing the main modes of variation. As a necessary step
to developing our methodology, we prove results on the existence and
boundedness of optimal multitransport maps. These are of independent interest
in the theory of transport of Gaussian processes. The transportation ANOVA and
PCA are illustrated on a variety of simulated and real examples.
