  We consider statistical models where functional data are artificially
contaminated by independent Wiener processes in order to satisfy privacy
constraints. We show that the corrupted observations have a Wiener density
which determines the distribution of the original functional random variables,
masked near the origin, uniquely, and we construct a nonparametric estimator of
that density. We derive an upper bound for its mean integrated squared error
which has a polynomial convergence rate, and we establish an asymptotic lower
bound on the minimax convergence rates which is close to the rate attained by
our estimator. Our estimator requires the choice of a basis and of two
smoothing parameters. We propose data-driven ways of choosing them and prove
that the asymptotic quality of our estimator is not significantly affected by
the empirical parameter selection. We examine the numerical performance of our
method via simulated examples.
