  There have been recently several works studying the regularized stochastic
heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq
3$ as the smoothing parameter is switched off, but most of the results did not
hold in the full temperature regions where they should. Inspired by martingale
techniques coming from the directed polymers literature, we first extend the
law of large numbers for SHE obtained in [MSZ16] to the full weak disorder
region of the associated polymer model and to more general initial conditions.
We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation
studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with
multidimensional convergence and general initial conditions for the KPZ
equation (and SHE), which were not proven before. To do so, we rely on a
martingale CLT combined with a refinement of the local limit theorem for
polymers.
