  We prove that the subquartic wave equation on the three dimensional ball
$\Theta$, with Dirichlet boundary conditions admits global strong solutions for
a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$.
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.
