Weak coupling limit of KPZ with rougher than white noise

TL;DR

This paper investigates the KPZ equation with noise rougher than white noise, showing that under weak coupling, the solutions converge to a Gaussian limit different from the linear solution, revealing new behaviors in the subcritical regime.

Contribution

It demonstrates the convergence of renormalised solutions of KPZ with very rough noise to a distinct Gaussian limit, extending understanding of the weak coupling regime.

Findings

Solutions converge to a Gaussian limit different from the linear solution.

The effect occurs in the subcritical regime with noise rougher than white.

Regularity structures are used to establish convergence.

Abstract

We consider the KPZ equation in $1$ spatial dimension with noise that is rougher than white by an exponent $\gamma>1/4$. Under a weak coupling limit, formally removing the nonlinearity from the equation, we show using regularity structures that the renormalised solutions converge to a Gaussian limit that is different from the solution of the linear part of the equation. The regime of this effect has a nontrivial overlap with the subcritical regime $\gamma<1/2$.