It\^o perspective on variance renormalisation

TL;DR

This paper demonstrates that solutions to a nonlinear stochastic heat equation with mollified noise converge to a limiting equation with a renormalized noise term, revealing a nontrivial scaling exponent in a supercritical setting.

Contribution

It provides a rigorous analysis of variance renormalization for the stochastic heat equation with supercritical noise, establishing convergence and identifying the renormalized limit.

Findings

Convergence of Itô solutions as mollification vanishes

Identification of the limiting equation with renormalized noise

The exponent 3/4 is critical for the limit in supercritical noise

Abstract

We show that the It\^o solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- \Delta u^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla \xi_\varepsilon, $$ where $ \xi_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $\xi$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)\xi$ with a constant $c>0$. Since the noise $\nabla\xi$ 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.