Moment-based approach for two erratic KPZ scaling limits

TL;DR

This paper presents a moment-based approach to analyze KPZ scaling limits, providing simpler proofs for existing results and addressing challenges in traditional methods for stochastic heat equations.

Contribution

It offers a new proof technique for KPZ-related results using moments, simplifying analysis and extending understanding in weaker topologies.

Findings

Recovered a variance blowup result for KPZ in weaker topology

Reproduced KPZ scaling limit for random walks in random environments

Provided a simpler proof avoiding complex traditional methods

Abstract

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.