The 2d random heat equation and the 2d KPZ equation with a general Gaussian noise

and the related KPZ equation



in the critical dimension d = 2 where V is a Gaussian random potential and β is the noise strength. We will focus on the case where the potential is not white in time and study the large-scale fluctuations of u(t, x) and h(t, x). We show that after renormalizing, the fluctuations converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. We discuss our main tools, a specific Markov chain on the space of paths and an extension of the Kallianpur-Robbins law to a specific regenerative process. Time permitting, we will also discuss possible future directions.