The two-dimensional KPZ equation in the entire subcritical regime

We consider the KPZ equation in space dimension 2 driven by space-time white noise. We showed in previous work that if the noise is mollified in space on scale $\epsilon$ and its strength is scaled as $\hat\beta / \sqrt{|\log \epsilon|}$, then a transition occurs with explicit critical point $\hat\beta_c = \sqrt{2\pi}$. Recently Chatterjee and Dunlap showed that the solution admits subsequential scaling limits as $\epsilon \downarrow 0$, for sufficiently small $\hat\beta$. We prove here that the limit exists in the entire subcritical regime $\hat\beta \in (0, \hat\beta_c)$ and we identify it as the solution of an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations. The same result holds for the directed polymer model in random environment in space dimension 2.