  We consider the energy-critical defocusing nonlinear wave equation (NLW) on
$\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and
uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times
H^{s-1}(\mathbb{R}^d)$, with $0< s\leq 1$ if $d=4$, and $0\leq s\leq 1$ if
$d=5$. The randomization we consider is naturally associated with the Wiener
decomposition and with modulation spaces. The proof is based on a probabilistic
perturbation theory. Under some additional assumptions, for $d=4$, we also
prove the probabilistic continuous dependence of the flow with respect to the
initial data (in the sense proposed by Burq and Tzvetkov).
