  The asymptotic behavior of a class of stochastic reaction-diffusion-advection
equations in the plane is studied. We show that as the divergence-free
advection term becomes larger and larger, the solutions of such equations
converge to the solution of a suitable stochastic PDE defined on the graph
associated with the Hamiltonian. Firstly, we deal with the case that the
stochastic perturbation is given by a singular spatially homogeneous Wiener
process taking values in the space of Schwartz distributions. As in previous
works, we assume here that the derivative of the period of the motion on the
level sets of the Hamiltonian does not vanish. Then, in the second part,
without assuming this condition on the derivative of the period, we study a
weaker type of convergence for the solutions of a suitable class of linear
SPDEs.
