Renormalisation in the presence of variance blowup

TL;DR

This paper investigates the renormalization of stochastic PDEs and SDEs driven by smoothed noise, showing convergence to equations driven by white noise under specific scaling limits, despite variance blowup issues.

Contribution

It introduces new scaling limits for KPZ and fractional Brownian motion driven equations, demonstrating convergence to white noise driven equations despite variance blowup.

Findings

Solutions converge to KPZ with white noise as smoothing scale vanishes.

Solutions converge to SDE driven by white noise for Hurst < 1/4.

Different proof techniques are used for the two convergence results.

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.