Abstract. We consider the Cauchy problem of the cubic nonlinear Schrö-dinger equation (NLS) : i∂ t u + Δu = ±|u| 2 u on R d , d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity. More precisely, given a function on R d , we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scalingcritical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove 'conditional' almost sure global well-posedness for d = 4 in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when d = 4, we show that conditional almost sure global wellposedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schrödinger equation (NLS) with initial data below L 2 . In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local wellposedness of NLS almost surely for the initial data in the support of the canonical Gaussian measures on H s (T) for each s > − 1 3 , and global well-posedness for each s > − 1 12 .

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a timedependent renormalization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW. 2010 Mathematics Subject Classification. 35L71, 60H15. Key words and phrases. stochastic nonlinear wave equation; nonlinear wave equation; renormalization; Wick ordering; Hermite polynomial; white noise. 1 Namely, replace βn in (1.3) by the sum of two independent Brownian motions, one forward in time on T 2 × [0, ∞) and the other backward in time T 2 × (−∞, 0], both starting at t = 0.