Itô perspective on variance renormalisation

Abstract

We show that the Itô solutions of the nonlinear stochastic heat equation $${\partial }_t{u}^{\epsilon }-\hat{I}”u^{\epsilon }={\epsilon }^{3/4}g(u^{\epsilon })\nabla \hat{I}{¾}_{\epsilon },$$ where $\hat{I}{¾}_{\epsilon }$ denotes the mollification in space at scale $\epsilon >0$ of a space-time white noise $\hat{I}¾$, converge in law, as $\epsilon \to 0$, to the solution of the stochastic heat equation with right-hand side $c{g}^{\prime }g(u)\hat{I}¾$ with a constant $c>0$. Since the noise $\nabla \hat{I}¾$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.

31 pages