Law of large numbers and fluctuations in the sub-critical and [formula omitted] regions for SHE and KPZ equation in dimension [formula omitted

Abstract

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) equation in dimension d≥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 Mukherjee et al. (2016) 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 Gu et al. (2018), Magnen and Unterberger (2018), Dunlap et al. (2020) to the full L2-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.