Many researchers in dispersive partial differential equations have recently examined whether blow-up behaviour, such as the norm-inflation described above, occurs for generic or only exceptional sets of rough initial data. To quantify this, one is quickly lead to random initial data. Indeed, one natural form of rough initial **data** is pu 0`f ω 0 , u 1`f ω 1 q, where the functions pu 0 , u 1 q P 9 H 1ˆL2 are regular and deterministic, while the functions pf ω 0 , f ω 1 q P H sˆH s´1 are rough and random. An analogue of Theorem 1.1 in this case would imply the stability of the scattering mechanism under a perturbation by noise. The literature on random dispersive partial differential equations is vast. We refer the interested reader to the survey [6], and mention the related works [2,4,8,9,11,12,14,15,16,38,39,41,42,44]. In the following discussion, we focus on the Wiener randomization [3,38] of a function f P H s pR d q. Let ϕ P C 8 pR d q be a smooth and symmetric function satisfying ϕ| r´3{8,3{8s d " 1, ϕ| R d zr´5{8,5{8s d " 0, and ř kPZ d ϕpξ´kq " 1 for all ξ P R d . We then define the associated operator P k by y P k f pξq :" ϕpξ´kq p f pξq .