| Abstract: |
| In this talk we will see 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 Hairer (2025) for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques. |
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