Random initial conditions for semi-linear PDEs

We analyze the effect of random initial conditions on the local well--posedness of semi--linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework. In particular, in some cases stochastic initial conditions extend the validity of the fixed--point argument to larger spaces than deterministic initial conditions would allow, but it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structures present in the equation. We also give a specific example where the level of regularity for the fixed--point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus critically cannot be reached even by random initial conditions. The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub--critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.