The analysis of solutions to some stochastic dissipative PDE (notably, the
Navier-Stokes equations) for which well-posedness is an open problem can be
explored using the idea of Markov solutions. Under appropriate assumptions
on the driving noise such solutions have some remarkable properties, such
as regularity (with respect to initial condition) and unique ergodicity.
Such properties seem to be closely related to the topology where the
equations are locally solvable.
It turns out that Markov solutions essentially provide a useful framework to
clearly state and recover several properties of solutions, such as convergence
to the invariant measure (including the case of mild degeneracy of the noise)
or stability with respect to small parameters.
We shall discuss some examples that try to clarify the whole picture.