Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on Rd, d=4 and 5

We consider the energy-critical defocusing nonlinear wave equation (NLW) on R-d, d = 4; 5. We prove almost sure global existence and uniqueness for NLWwith rough random initial data in H-s. (R-d) x Hs(-1) (R-d) with 0 < s <= 1 if d = 4, and 0 <= s <= 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 on the initial data (in the sense proposed by Burq and Tzvetkov [19]).