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

Probabilistic well-posedness for the cubic wave equation