An asymptotic approach to SPDEs on graphs

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