Random data Cauchy theory for supercritical wave equations II : A global existence result

We prove that the subquartic wave equation on the three dimensional ball $\mathrm{\Theta }$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in ${\cap }_{s<1\mathrm{/}2}{H}^s(\mathrm{\Theta })$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.