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 $\epsilon \ll 1$ and multiplied by ${\epsilon }^{3/4}$ then, as $\epsilon \to 0$, 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 $\epsilon \ll 1$ and multiplied by ${\epsilon }^{1/4-H}$ then, as $\epsilon \to 0$, 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.

Accepted version

Cited by in corpus (1)

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