Moment-based approach for two erratic KPZ scaling limits
Year: 2025
Type: article
Abstract: A recent paper of Tsai [Tsa24] 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 [Hai24], albeit in a much weaker topology. The second one recovers a KPZ scaling limit result related to random walks in random environments from [Par24], 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.
DOI: 10.1214/25-ejp1323
Author: Shalin Parekh
Institution: University of Maryland, College Park
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W4409305380