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 Theta, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in boolean AND(s<1/2)(H-s(Theta) x Hs-1(Theta)). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2,3] on the non linear Schrodinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.