I will discuss the difficulties which arise when one considers the
long time behavior of a stochastically forced PDE. I will try to
highlight that there are different cases which require very different
ideas. Some cases can be seen as extensions of what is done in finite
dimensions, others require new tools and ideas. I will concentrate on
the case of degenerately forced SPDEs. I will describe an extension of
Hormander's "sum of squares theorem" to hypo-elliptic operators in
infinite dimensions. I will discuss the concert examples of the 2D
Navier Stokes equations on the torus and sphere as well as a class of
reaction diffusion equations. In these contexts the discussion will
center on the transfer of randomness between scales.