  The purpose of this article is to introduce for dispersive partial
differential equations with random initial data, the notion of well-posedness
(in the Hadamard-probabilistic sense). We restrict the study to one of the
simplest examples of such equations: the periodic cubic semi-linear wave
equation. Our contributions in this work are twofold: first we break the
algebraic rigidity involved in previous works and allow much more general
randomizations (general infinite product measures v.s. Gibbs measures), and
second, we show that the flow that we are able to construct enjoys very nice
dynamical properties, including a new notion of probabilistic continuity.
