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The asymptotic behavior of a class of stochastic reaction-diffusion-advectionequations in the plane is studied. We show that as the divergence-freeadvection term becomes larger and larger, the solutions of such equationsconverge to the solution of a suitable stochastic PDE defined on the graphassociated with the Hamiltonian. Firstly, we deal with the case that thestochastic perturbation is given by a singular spatially homogeneous Wienerprocess taking values in the space of Schwartz distributions. As in previousworks, we assume here that the derivative of the period of the motion on thelevel sets of the Hamiltonian does not vanish. Then, in the second part,without assuming this condition on the derivative of the period, we study aweaker type of convergence for the solutions of a suitable class of linearSPDEs.

Incompressible viscous fluids in $\mathbb{R}^2$ and SPDEs on graphs, in presence of fast advection and non smooth noise