The Scaling Limit of the KPZ Equation in Space Dimension 3 and Higher

Abstract

We study in the present article the Kardar–Parisi–Zhang (KPZ) equation $$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$ in $$d\ge 3$$ dimensions in the perturbative regime, i.e. for $$\lambda >0$$ small enough and a smooth, bounded, integrable initial condition $$h_0=h(t=0,\cdot )$$ . The forcing term $$\eta $$ in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85–121, 1994). Standard large deviation estimates for $$\eta $$ make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards–Wilkinson model ( $$\lambda =0$$ ) with renormalized coefficients $$\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)$$ .