Renormalisation in the presence of variance blowup

Abstract

We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{3/4}$ then, as $\varepsilon \to 0$, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise. In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter $H < 1/4$, smoothened out at scale $\varepsilon \ll 1$ and multiplied by $\varepsilon^{1/4-H}$ then, as $\varepsilon \to 0$, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially.