Random data Cauchy theory for supercritical wave equations I: Local theory

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1\mathrm{/}2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in $H^s(M)$, where $s\ge 1\mathrm{/}4$ in the case of a boundary less manifold and $s\ge 8\mathrm{/}21$ in the case of a manifold with boundary.