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

We consider the stochastic PDE: ${\mathrm{\partial }}_{t}u(t,x)=\frac{1}{2}\mathrm{\Delta }u(t,x)+\beta u(t,x)V(t,x),$ 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.