In the classical well-posedness theory for nonlinear dispersive and hyper
bolic equations the aim is to construct unique strong solutions for all in
itial data belonging to a certain function space such as the L^2-based Sob
olev spaces. However\, at low regularities ill-posedness phenomena usually
tend to occur. In practice one is often interested in the typical behavio
r of solutions and may be content to neglect certain pathological behavior
s leading to ill-posedness results. This concept may be formalized by cons
idering random initial data and by trying to construct in an almost sure m
anner strong local-in-time or even global-in-time solutions. Such an appro
ach sometimes allows to go beyond certain deterministic regularity thresho
lds.

I will begin this talk with a general introduction to the
study of nonlinear dispersive and hyperbolic equations for random initial
data. Afterwards I will present an almost sure global existence and scatte
ring result for the 4D energy-critical nonlinear wave equation for scaling
super-critical random data in the radial case.

This talk is ba
sed on joint works with Ben Dodson and Dana Mendelson.