Title: Random data wave equations,
Abstract: Nowadays we have many methods allowing to exploit the regularising properties
of the linear part of a nonlinear dispersive equation (such as the KdV
equation, the nonlinear wave or the nonlinear Schroedinger equations) in order
to prove well-posedness in low regularity Sobolev spaces. By well-posedness in
low regularity Sobolev spaces we mean that less regularity than the one imposed
by the energy methods is required (the energy methods do not exploit the
dispersive properties of the linear part of the equation). In many cases these
methods to prove well-posedness in low regularity Sobolev spaces lead to
optimal results in terms of the regularity of the initial data. By optimal we
mean that if one requires slightly less regularity then the corresponding
Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev
spaces in which these ill-posedness results hold spaces of supercritical
regularity.
More recently, methods to prove probabilistic well-posedness in Sobolev
spaces of supercritical regularity were developed. More precisely, by
probabilistic well-posedness we mean that one endows the corresponding Sobolev
space of supercritical regularity with a non degenerate probability measure and
then one shows that almost surely with respect to this measure one can define a
(unique) global flow. However, in most of the cases when the methods to prove
probabilistic well-posedness apply, there is no information about the measure
transported by the flow. Very recently, a method to prove that the transported
measure is absolutely continuous with respect to the initial measure was
developed. In such a situation, we have a measure which is quasi-invariant
under the corresponding flow.
The aim of these lectures is to present all of the above described
developments in the context of the nonlinear wave equation.