KPZ equation with fractional derivatives of white noise

TL;DR

This paper investigates the well-posedness of the KPZ equation driven by fractional derivatives of white noise, revealing that the existing regularity structures theory applies only for certain noise roughness levels, with renormalization issues arising at higher roughness.

Contribution

It extends the analysis of the KPZ equation to fractional derivatives of white noise, identifying the threshold for well-posedness and renormalization applicability.

Findings

Regularity structures theory applies if c1<1/2.

Renormalization is well-posed only if c1<1/4.

The rougher the noise, the more challenging the analysis.

Abstract

In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order $\gamma>0$ in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer [3] as an application of the theory of regularity structures. Our aim is to see to what extent his theory works if noises become rougher. We can expect that his theory works if and only if $\gamma<1/2$. However, we show that the renormalization like "$(\partial_x h)^2-\infty$" is well-posed only if $\gamma<1/4$.