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We consider the energy-critical defocusing nonlinear wave equation (NLW) on$\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence anduniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\timesH^{s-1}(\mathbb{R}^d)$, 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 Wienerdecomposition and with modulation spaces. The proof is based on a probabilisticperturbation theory. Under some additional assumptions, for $d=4$, we alsoprove the probabilistic continuous dependence of the flow with respect to theinitial data (in the sense proposed by Burq and Tzvetkov).

Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$