Moment-based approach for two erratic KPZ scaling limits

A recent paper of Tsai shows how the first few moments of a stochastic flow in the space of measures can completely determine its law. Here we give another proof of this result for the particular case of the one-dimensional multiplicative stochastic heat equation (mSHE), and then we investigate two corollaries. The first one recovers a recent result of Hairer on a ``variance blowup" problem related to the KPZ equation , albeit in a much weaker topology. The second one recovers a KPZ scaling limit result related to random walks in random environments, but in a weaker topology. In these two problems, we furthermore explain why it is hard to directly use the martingale characterization of the mSHE, the chaos expansion, or other known methods. Using the moment-based approach avoids technicalities, leading to a short proof.