Law of large numbers and fluctuations in the sub-critical and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si3.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> regions for SHE and KPZ equation in dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e30" altimg="si4.svg"><mml:mrow><mml:mi>d</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="…

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) in dimension d≥3 as smoothing parameter is switched off, but most of results did not hold full temperature regions where they should. Inspired by martingale techniques coming from directed polymers literature, we first extend law large numbers for SHE obtained Mukherjee et al. (2016) to weak disorder region associated polymer model more general initial conditions. We further Edwards–Wilkinson regime KPZ studied Gu (2018), Magnen Unterberger Dunlap (2020) L2-region, along with multidimensional convergence conditions (and SHE), which were proven before. To do so, rely on a CLT combined refinement local limit theorem polymers.