The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations

We consider the stochastic PDE: ₜu (t, x) =12 u (t, x) +u (t, x) V (t, x), in dimension d=2, where the potential V is the space and time mollification of the two-dimensional space-time white noise. We show that after renormalizing, the fluctuations of the solution converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. Our main tool is a Markov chain on the space of paths which we use to establish an extension of the Kallianpur-Robbins law to a specific regenerative process.